Abstract
The purpose of this paper is to give a direct proof of an eigenfunction expansion formula for one-dimensional two-state quantum walks, which is an analog of that for Sturm–Liouville operators due to Weyl, Stone, Titchmarsh, and Kodaira. In the context of the theory of CMV matrices, it had been already established by Gesztesy–Zinchenko. Our approach is restricted to the class of quantum walks mentioned above, whereas it is direct and it gives some important properties of Green functions. The properties given here enable us to give a concrete formula for a positive-matrix-valued measure, which gives directly the spectral measure, in a simplest case of the so-called two-phase model.
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1 Introduction
The quantum walks are certain unitary operators, and they are sometimes regarded as quantum counterparts of the classical random walks. The homogeneous two-state quantum walks (in one dimension with a constant coin matrix) are well understood fundamental models (see, for example, [6, 9, 25]), and recently the scattering-theoretical aspects, as a perturbation of homogeneous walks, are intensively investigated (see [15,16,17, 20, 21]). The Schrödinger operators in one dimension are often called the Sturm–Liouville operators and they are well studied. Thus, it would be rather natural to understand resemblance between one-dimensional quantum walks and Sturm–Liouville operators. The purpose of the present paper is to give a proof of an eigenfunction expansion formula for one-dimensional two-state quantum walks which is analogous to classical formulas of Weyl [27, 28], Stone [24], Titchmarsh [26], and Kodaira [10] for Sturm–Liouville operators. The theory of eigenfunction expansions for Sturm–Liouville operators are discussed, for example, in [12, 14, 18, 19] and a short review can be found in [23]. Probabilistic aspects of one-dimensional quantum walks are also intensively investigated. The notion of transfer matrix is introduced in [11] to construct stationary measures from eigenfunctions for quantum walks, and it is suitable for our analysis. Then, our basic idea in this paper is to use the transfer matrix to develop a theory analogous to that for Sturm–Liouville operators.
Before going to explain our setting-up, we should mention about the work by Gesztesy–Zinchenko [8] on Weyl–Titchmarsh theory for CMV matrices with Verblunsky coefficients in the unit disk. The notion of CMV matrices has been introduced by Cantero–Moral–Velázquez [2] and has further developed and deepened by Simon [22, 23]. The one-dimensional two-state quantum walks are special CMV matrices and the theory of CMV matrices applied to this class of quantum walks in [1, 5] and other works. Therefore, many of our results in this paper are essentially contained in [8]. However, since our presentations and proofs are direct without using the theory of CMV matrices, and formulas are given in usual representations of unitary evolutions for quantum walks. Although our approach only works for the class of quantum walks mentioned above, the setting of our presentation could have advantageous aspect when the quantum walks are applied and used in areas different from pure mathematics such as information science or quantum physics. Furthermore, it seems that a property of the Green function, Theorem 1.4 below, is new, and it can be used to give a concrete formula, Theorem 5.7, to compute the positive-matrix-valued measure, which gives directly the spectral resolution, for a certain special simplest case of the so-called two-phase model [3].
Now, let us prepare notation to mention some of results in the paper. All the inner products in the paper are complex linear in the first variable and anti-complex linear in the second. We denote the standard Hermitian inner product of the two-dimensional complex vector space \({\mathbb{C}}^{2}\) by \(\langle \,\cdot ,\cdot \,\rangle _{{\mathbb{C}}^{2}}\) and the standard orthonormal basis of \({\mathbb{C}}^{2}\) by \(\{e_{L}, e_{R}\}\)
The orthogonal projection onto the one-dimensional subspaces, \({\mathbb{C}}e_{L},\) \({\mathbb{C}}e_{R},\) are denoted by \(\pi _{L},\) \(\pi _{R}.\) In general, the set of maps from a set X to another set Y is denoted by \({\mathrm{Map}}\,(X,Y).\) We fix \({\mathcal{C}} \in {\mathrm{Map}}\,({\mathbb{Z}},{\mathrm{U}}(2)),\) where \({\mathrm{U}}(2)\) is the group of unitary \(2 \times 2\) matrices, and define a linear map
by the following formula:
where \(\Psi \in {\mathrm{Map}}\,({\mathbb{Z}},{\mathbb{C}}^{2}),\) \(n \in {\mathbb{Z}}.\) Let \(\ell ^{2}({\mathbb{Z}},{\mathbb{C}}^{2})\) be the Hilbert space of \(\ell ^{2}\)-functions whose inner product is given by
The linear map \(U({\mathcal{C}})\) defined in (2) becomes a unitary operator on \(\ell ^{2}({\mathbb{Z}},{\mathbb{C}}^{2})\) when it is restricted to \(\ell ^{2}({\mathbb{Z}},{\mathbb{C}}^{2}),\) and it preserves the space \(C_{0}({\mathbb{Z}},{\mathbb{C}}^{2})\) of finitely supported \({\mathbb{C}}^{2}\)-valued functions. In this paper, we call the linear map defined in (1), (2) the quantum walk with the coin matrix \({\mathcal{C}}:{\mathbb{Z}} \rightarrow {\mathrm{U}}(2).\) We write
Throughout this paper, we assume the following:
Under the assumption (4), the unitarity of the matrix \({\mathcal{C}}(n)\) causes \(d_{n} \ne 0\) for any \(n \in {\mathbb{Z}}.\)
Theorem 1.1
[11, 13, 17] Suppose that the coin matrix \({\mathcal{C}}\) satisfies the assumption (4). For any \(n \in {\mathbb{Z}}\) and any \(\lambda \in {\mathbb{C}} {\setminus } \{0\},\) we define a \(2 \times 2\) matrix \(T_{\lambda }(n)\) by
and a \(2 \times 2\) matrix \(F_{\lambda }(n)\) by
For any \(u \in {\mathbb{C}}^{2},\) we define \(\Phi _{\lambda }(u) \in {\mathrm{Map}}\,({\mathbb{Z}},{\mathbb{C}}^{2})\) by
Then, the map \(\Phi _{\lambda } :{\mathbb{C}}^{2} \rightarrow {\mathrm{Map}}\,({\mathbb{Z}},{\mathbb{C}}^{2})\) is injective and the eigenspace \({\mathcal{M}}^{\lambda }\) of \(U({\mathcal{C}})\) in \({\mathrm{Map}}\,({\mathbb{Z}},{\mathbb{C}}^{2})\) with an eigenvalue \(\lambda \in {\mathbb{C}} {\setminus } \{0\}\) coincides with the image of \(\Phi _{\lambda }.\) Hence, \(\dim {\mathcal{M}}^{\lambda }=2\) for each such \(\lambda .\) Furthermore, we have \(\dim {\mathcal{M}}^{\lambda } \cap \ell ^{2}({\mathbb{Z}},{\mathbb{C}}^{2}) \le 1\) for any \(\lambda \in S^{1}.\)
The matrix \(T_{\lambda }(n)\) is called the transfer matrix,Footnote 1 and it is useful to describe various functions and quantities related to the quantum walk \(U({\mathcal{C}}).\) For example, the Green function can be expressed in terms of the transfer matrix. To be more precise, let
be the resolvent of the restriction U of \(U({\mathcal{C}})\) to \(\ell ^{2}({\mathbb{Z}},{\mathbb{C}}^{2}),\) where \(\sigma (U)\) denotes the spectrum of the operator U. For any \(u \in {\mathbb{C}}^{2}\) and \(k \in {\mathbb{Z}},\) we define a function \(\delta _{k} \otimes u \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2})\) by
The Green function \(R_{\lambda } \in {\mathrm{Map}}\,({\mathbb{Z}}^{2}, {\mathrm{M}}_{2}({\mathbb{C}})),\) where \({\mathrm{M}}_{2}({\mathbb{C}})\) denotes the space of complex \(2 \times 2\) matrices, is defined as
Then, the Green function \(R_{\lambda }(n,m)\) is expressed in terms of the matrix \(F_{\lambda }(n)\) as in the following.
Theorem 1.2
We define \({\mathrm{z}}_{L},{\mathrm{z}}_{R} \in {\mathrm{Map}}\,({\mathbb{Z}},{\mathrm{M}}_{2}({\mathbb{C}}))\) by
We set \({\mathrm{x}}_{0}(\lambda )=R_{\lambda }(0,0).\) Then, we have
and for each \(n \in {\mathbb{Z}},\) we have
As in Theorem 1.2, the matrix-valued holomorphic function \({\mathrm{x}}_{0}(\lambda )=R_{\lambda }(0,0)\) plays one of the central roles in the present paper. Therefore, it is important to develop methods to compute \({\mathrm{x}}_{0},\) one of which is given by the following two theorems.
Theorem 1.3
Let \(\lambda \in {\mathbb{C}} {\setminus } (\{0\} \cup S^{1}).\) Let \(A_{L}(\lambda )\) (resp. \(A_{R}(\lambda ))\) be the vector subspace in \({\mathbb{C}}^{2}\) consisting of all vectors \(w \in {\mathbb{C}}^{2}\) satisfying
Then, we have \(\dim A_{L}(\lambda )=\dim A_{R}(\lambda )=1.\) In particular, we have
for any \(\lambda \in {\mathbb{C}} {\setminus } (\{0\} \cup S^{1})\) and \(n,m \in {\mathbb{Z}}\) with \(n \ne m.\)
For \(\lambda \in {\mathbb{C}} {\setminus } (\{0\} \cup S^{1}),\) let \({\mathrm{v}}_{\pm }(\lambda )\) be unit vectors satisfying
The existence of these unit vectors is assured by Theorem 1.3.
Theorem 1.4
For \(\lambda \in {\mathbb{C}} {\setminus } (\{0\} \cup S^{1})\) the unit vectors \({\mathrm{v}}_{+}(\lambda ), {\mathrm{v}}_{-}(\lambda )\) are linearly independent. The matrix-valued holomorphic function \({\mathrm{x}}_{0}(\lambda )=R_{\lambda }(0,0)\) is given by
where \({\mathrm{v}}_{\pm }(\lambda )^{\perp }\) denotes any unit vector in \({\mathbb{C}}^{2}\) perpendicular to \({\mathrm{v}}_{\pm }(\lambda ),\) respectively.
The eigenfunction expansion theorem due to Weyl [27, 28], Stone [24], Titchmarsh [26], and Kodaira [10]Footnote 2 is regarded as an inversion formula for a generalized Fourier transform defined by eigenfunctions for Sturm–Liouville operators. Let us state an eigenfunction expansion formula for the quantum walk \(U({\mathcal{C}})\) defined by the coin matrix \({\mathcal{C}}\) satisfying (4). For any \(f \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2}),\) we define a function \({\mathcal{F}}_{{\mathcal{C}}}[f]\) on \({\mathbb{C}} {\setminus } \{0\}\) by
The sum in (16) is finite, because the support of f is finite. Therefore, the function \({\mathcal{F}}_{{\mathcal{C}}}[f](\lambda )\) is a Laurent polynomial in \(\lambda \in {\mathbb{C}} {\setminus } \{0\}.\) We call \({\mathcal{F}}_{{\mathcal{C}}}[f]=\widehat{f}^{{\mathcal{C}}}\) the QW-Fourier transform of f.
Theorem 1.5
There exists a positive-matrix-valued measure \(\Sigma\) on \(S^{1}\), such that we have the following.
-
(1)
The resolvent \(R(\lambda )\) is written as
$$\begin{aligned} \langle \,R(\lambda )f,g \,\rangle =\int _{S^{1}} \frac{1}{\zeta -\lambda } \langle \,{\mathrm{d}}\Sigma (\zeta ){\mathcal{F}}_{{\mathcal{C}}}[f](\zeta ),{\mathcal{F}}_{{\mathcal{C}}}[g](\zeta ) \,\rangle _{{\mathbb{C}}^{2}}. \end{aligned}$$(17)The positive-matrix-valued measure \(\Sigma\) satisfying (17) is unique.
-
(2)
For any \(f,g \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2}),\) we have
$$\begin{aligned} \langle \,f,g \,\rangle =\int _{S^{1}} \langle \,{\mathrm{d}}\Sigma (\zeta ){\mathcal{F}}_{{\mathcal{C}}}[f](\zeta ),{\mathcal{F}}_{{\mathcal{C}}}[g](\zeta ) \,\rangle _{{\mathbb{C}}^{2}}. \end{aligned}$$(18) -
(3)
Let
$$\begin{aligned} U=\int _{S^{1}} \lambda \,{\mathrm{d}}E(\lambda ) \end{aligned}$$be the spectral resolution of the unitary operator U, where E is a projection-valued measure on \(S^{1}.\) Then, for each Borel set A in \(S^{1},\) the projection E(A) on \(\ell ^{2}({\mathbb{Z}},{\mathbb{C}}^{2})\) is written as
$$\begin{aligned}{}[E(A)f](n)=\int _{A} F_{\zeta }(n) {\mathrm{d}}\Sigma (\zeta ) {\mathcal{F}}_{{\mathcal{C}}}[f](\zeta ),\quad f \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2}). \end{aligned}$$(19)In particular, the following inversion formula holds for \(f \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2}):\)
$$\begin{aligned} f(n)=\int _{S^{1}} F_{\zeta }(n) {\mathrm{d}}\Sigma (\zeta ) {\mathcal{F}}_{{\mathcal{C}}}[f](\zeta ). \end{aligned}$$(20)
Corollary 1.6
The following holds.
-
(1)
The spectrum \(\sigma (U)\) coincides with the support of \(\Sigma .\)
-
(2)
\(\lambda \in S^{1}\) is an eigenvalue of U if and only if \(\Sigma (\{\lambda \}) \ne 0.\) When \(\lambda\) is an eigenvalue of U, the projection \(E(\{\lambda \})\) onto the eigenspace of \(\lambda\) is given by
$$\begin{aligned}{}[E(\{\lambda \})f](n)=F_{\lambda }(n)\Sigma (\{\lambda \}) {\mathcal{F}}_{{\mathcal{C}}}[f](\lambda ). \end{aligned}$$
We refer the readers to [7, 8] for properties of positive-matrix-valued measures. We note that the matrix-valued function \({\mathrm{x}}_{0}(\lambda )\) is not an m-Carathéodory function in the sense of [22], because our operator is unitary. Instead, we use the matrix
which is indeed an m-Carathéodory function. The positive-matrix-valued measure \(\Sigma\) is then a boundary value of the function \({\mathrm{x}}(\lambda )\) in the sense that \(\Sigma\) satisfies
and \(\Sigma\) is characterized as
Let \(C(S^{1}\!,{\mathbb{C}}^{2})\) be the space of continuous \({\mathbb{C}}^{2}\)-valued functions on \(S^{1}.\) For any \(k, l \in C(S^{1},{\mathbb{C}}^{2}),\) we define
This is a positive semi-definite Hermitian sesquilinear form on \(C(S^{1}\!,{\mathbb{C}}^{2})\), and hence, it defines an inner product on the quotient space \(C(S^{1},{\mathbb{C}}^{2})/N\) of \(C(S^{1},{\mathbb{C}}^{2})\) by the subspace \(N=\{k \in C(S^{1}\!,{\mathbb{C}}^{2}) \mid \Vert k\Vert _{\Sigma }=0\},\) where we set
We denote \(L^{2}(S^{1}\!,{\mathbb{C}}^{2})_{\Sigma }\) the completion of \(C(S^{1}\!,{\mathbb{C}}^{2})/N\) by the norm on \(C(S^{1},{\mathbb{C}}^{2})/N\) naturally induced by (25). The QW-Fourier transform (16) induces a map from \(C_{0}({\mathbb{Z}},{\mathbb{C}}^{2})\) to \(L^{2}(S^{1},{\mathbb{C}}^{2})_{\Sigma }\) which we denote \(\mathfrak {F}_{{\mathcal{C}}}.\)
Theorem 1.7
The map \(\mathfrak {F}_{{\mathcal{C}}}\) extends to a unitary operator from \(\ell ^{2}({\mathbb{Z}},{\mathbb{C}}^{2})\) to \(L^{2}(S^{1}\!, {\mathbb{C}}^{2})_{\Sigma }.\) The quantum walk U on \(\ell ^{2}({\mathbb{Z}},{\mathbb{C}}^{2})\) is unitarily equivalent to the operator defined by the multiplication by \(\lambda \in S^{1}\) on \(L^{2}(S^{1}\!, {\mathbb{C}}^{2})_{\Sigma },\) namely, we have
for any \(f \in \ell ^{2}({\mathbb{Z}},{\mathbb{C}}^{2}).\)
The organization of the paper is as follows. In Sect. 2, we solve two equations, an inhomogeneous eigenvalue equation and its conjugate. The definition of the QW-Fourier transform (16) comes from the fact that it gives a defect of the left-inverse of \(U({\mathcal{C}})-\lambda\) obtained by solving a conjugate equation to an inhomogeneous eigenvalue equation to be the right-inverse. See Theorem 2.6 for a precise statement. The solutions to these equations are used to prove Theorem 1.2 in Sect. 3. Some of the properties of the Green functions, such as Theorem 1.3, are proved also in Sect. 3. In Sect. 4, we give proofs of Theorems 1.5 and 1.7. We calculate the positive-matrix-valued measure \(\Sigma\) in two examples, homogeneous quantum walks, and a simplest case of the two-phase models in Sect. 5.
2 Inhomogeneous eigenvalue equations and its conjugate
Let \(f \in {\mathrm{Map}}\,({\mathbb{Z}},{\mathbb{C}}^{2}),\) \(\lambda \in {\mathbb{C}} {\setminus } \{0\}\) and \(w \in {\mathbb{C}}^{2}.\) We consider the following initial value problem:
Any \(\Psi (a)\) with a fixed integer \(a \in {\mathbb{Z}}\) can be chosen for an initial value, but we have chosen \(a=0\) for simplicity of notation. To prove Theorem 1.2, it is important to construct solutions to the problem (27) and its conjugate problem
where the map
is the extension of the formal adjoint operator (on \(C_{0}({\mathbb{Z}},{\mathbb{C}}^{2})\)) of \(U({\mathcal{C}})\) given by
A brief account on the formal adjoint operator for a linear map \(A :C_{0}({\mathbb{Z}},{\mathbb{C}}^{2}) \rightarrow {\mathrm{Map}}\,({\mathbb{Z}},{\mathbb{C}}^{2})\) is given in Appendix B. It is well known that \(U({\mathcal{C}})\) defines a unitary operator on \(\ell ^{2}({\mathbb{Z}},{\mathbb{C}}^{2}).\) This property comes from the following lemma.
Lemma 2.1
As linear maps on \({\mathrm{Map}}\,({\mathbb{Z}},{\mathbb{C}}^{2}),\) we have \(U({\mathcal{C}})^{*}=U({\mathcal{C}})^{-1}.\)
Proof
Let \(f \in {\mathrm{Map}}\,({\mathbb{Z}},{\mathbb{C}}^{2}).\) Then, we have
which shows the assertion. \(\square\)
In Proposition 2.3 below, we give formulas for the solutions to the initial value problems (27) and (28). Before proceed to Proposition 2.3 and its proof, we give recurrence equations equivalent to the equations in (27) and (28).
Lemma 2.2
The initial value problem (27) is equivalent to the equation
with \(\Psi (0)=w.\) The initial value problem (28) is equivalent to the equation
with \(\Psi (0)=w.\)
Proof
We suppose that \(\Psi \in {\mathrm{Map}}\,({\mathbb{Z}},{\mathbb{C}}^{2})\) is a solution to Eq. (27) with \(\Psi (0)=w \in {\mathbb{C}}^{2}.\) We set
so that Eq. (27) is written as
By the assumption (4), we can rewrite (34) as
Shifting the variable n in (34), we have
Solving the second equation of (36) in \(\psi _{R}(n+1)\), we have
Substituting (37) into the first equation of (35), we see
Equations (37) and (38) give Eq. (31). Next, we suppose that \(\Psi \in {\mathrm{Map}}\,({\mathbb{Z}},{\mathbb{C}}^{2})\) satisfies (31) and \(\Psi (0)=w.\) We write \(\Psi\) as in (33). Then, \(\psi _{L},\) \(\psi _{R}\) satisfy (37) and (38). Solving (37) in \(f_{R}(n+1)\) and substituting the result into (38), we obtain the first equation in (35). Shifting the variable n to \(n-1\) in (37), we have the second equation in (35). Therefore, \(\Psi\) satisfies (35). Since (35) is equivalent to the equation in (27), \(\Psi\) solves the initial value problem (27).
Next, we assume that \(\Psi \in {\mathrm{Map}}\,({\mathbb{Z}},{\mathbb{C}}^{2})\) solves the initial value problem (28), and we write \(\Psi\) as in (33). Equation (28) is equivalent to the equation
with \(\Psi (0)=w,\) which is written as
Shifting the variable n in (39), we see
Solving the first equation of (40) in \(\psi _{L}(n+1),\) we see
Substituting the second equation of (39) into (41) shows
Since the combination of the two equations, (42) and the second line of (39), is equivalent to Eq. (32), \(\Psi\) solves (32). Conversely, we suppose that \(\Psi \in {\mathrm{Map}}\,({\mathbb{Z}},{\mathbb{C}}^{2})\) satisfies (32) with \(\Psi (0)=w.\) Thus, \(\Psi\) satisfies (42) and the second line of (39). From these two equations, we have (41). Shifting the variable n in (41) to \(n-1\) and solving it in \(\psi _{L}(n-1)\), we have the first line of (39). Therefore, \(\Psi\) solves Eq. (28). \(\square\)
Lemma 2.2 is used to deduce the concrete formulas of the functions \(v_{\lambda }(n,m),\) \(w_{\lambda }^{o}(n,m)\) defined in (44), (47) below which give solutions to the problems (27), (28). Indeed, if we give an initial value \(\Psi (0)=w,\) then the function \(\Psi\) on the set of non-negative integers satisfying Eq. (32) is automatically determined. Shifting the variable n to \(n-1\) in (32) and solving it in \(\Psi (n-1),\) we see
Thus, once we give \(\Psi (0)=w,\) the function \(\Psi\) on the set of non-positive integers satisfying Eq. (43) is also automatically determined. Therefore, in principle, we can solve the Eq. (28). Similar discussion is also applicable for (27) using (31). The formulas of the solutions given in Proposition 2.3 are deduced from the concrete form of \(\Psi (n)\) obtained using Eqs. (31) and (32) for several small \(n \in {\mathbb{Z}}\) in the absolute value.
Proposition 2.3
-
(1)
We define a function \(v_{\lambda } \in {\mathrm{Map}}\,({\mathbb{Z}}^{2},{\mathrm{M}}_{2}({\mathbb{C}}))\) by the following formula:
$$\begin{aligned}&v_{\lambda }(n,0) = \left\{ \begin{array}{ll} \lambda ^{-1} F_{\lambda }(n) {\mathrm{z}}_{L}(0) &{}\quad (n \ge 1), \\ \lambda ^{-1}F_{\lambda }(n) {\mathrm{z}}_{R}(0) &{}\quad (n \le -1), \end{array}\right. \\ &v_{\lambda }(n,m) \\ &\quad = \left\{ \begin{array}{ll} \lambda ^{-1}F_{\lambda }(n)F_{\lambda }(m)^{-1}({\mathrm{z}}_{L}(m)-{\mathrm{z}}_{R}(m)) &{}\quad (1 \le m \le n-1), \\ -\lambda ^{-1} {\mathrm{z}}_{R}(n) &{}\quad (1 \le m =n), \\ \lambda ^{-1} F_{\lambda }(n) F_{\lambda }(m)^{-1} ({\mathrm{z}}_{R}(m) -{\mathrm{z}}_{L}(m)) &{}\quad (n+1 \le m \le -1), \\ -\lambda ^{-1} {\mathrm{z}}_{L}(n) &{}\quad (n=m \le -1), \\ 0 &{}\quad ({\mathrm{otherwise}}). \end{array}\right. \end{aligned}$$(44)Then, for each \(f \in {\mathrm{Map}}\,({\mathbb{Z}},{\mathbb{C}}^{2}),\) \(\lambda \in {\mathbb{C}} {\setminus } \{0\}\) and \(w \in {\mathbb{C}}^{2},\) Eq. (27) has a unique solution given by
$$\begin{aligned} \Psi =\Phi _{\lambda }(w)+V_{\lambda }f, \end{aligned}$$(45)where \(\Phi _{\lambda }(w)\) is defined in (7) and \(V_{\lambda }f\) is defined by
$$\begin{aligned} V_{\lambda }f(n)= \sum _{m \in {\mathbb{Z}}} v_{\lambda }(n,m)f(m). \end{aligned}$$(46) -
(2)
We define a function \(w_{\lambda }^{o} \in {\mathrm{Map}}\,({\mathbb{Z}}^{2},{\mathrm{M}}_{2}({\mathbb{C}}^{2}))\) by the following formula:
$$\begin{aligned}&w_{\lambda }^{o}(n,0) = \left\{ \begin{array}{ll} \overline{\lambda }^{-1}F_{1/\overline{\lambda }}(n){\mathrm{z}}_{R}(0)^{*} &{}\quad (n \ge 1), \\ \overline{\lambda }^{-1}F_{1/\overline{\lambda }}(n){\mathrm{z}}_{L}(0)^{*} &{}\quad (n \le -1), \end{array}\right. \\ &w_{\lambda }^{o}(n,m) \\ &\quad = \left\{ \begin{array}{ll} \overline{\lambda }^{-1}F_{1/\overline{\lambda }}(n)F_{1/\overline{\lambda }}(m)^{-1}({\mathrm{z}}_{R}(m)^{*} -{\mathrm{z}}_{L}(m)^{*}) &{}\quad (1 \le m \le n-1), \\ -\overline{\lambda }^{-1}{\mathrm{z}}_{L}(n)^{*} &{}\quad (1 \le m =n), \\ \overline{\lambda }^{-1} F_{1/\overline{\lambda }}(n)F_{1/\overline{\lambda }}(m)^{-1} ({\mathrm{z}}_{L}(m)^{*}-{\mathrm{z}}_{R}(m)^{*}) &{}\quad (n+1 \le m \le -1), \\ -\overline{\lambda }^{-1}{\mathrm{z}}_{R}(n)^{*} &{}\quad (n=m \le -1), \\ 0 &{} ({\mathrm{otherwise}}). \end{array}\right. \end{aligned}$$(47)Then, for each \(f \in {\mathrm{Map}}\,({\mathbb{Z}},{\mathbb{C}}^{2}),\) \(\lambda \in {\mathbb{C}} {\setminus } \{0\}\) and \(w \in {\mathbb{C}}^{2},\) Eq. (28) has a unique solution given by
$$\begin{aligned} \Psi =\Phi _{1/\overline{\lambda }}(w)+W_{\lambda }^{o}f, \end{aligned}$$(48)where \(W_{\lambda }^{o}f\) is defined by
$$\begin{aligned} W_{\lambda }^{o}f(n)= \sum _{m \in {\mathbb{Z}}} w_{\lambda }^{o}(n,m)f(m). \end{aligned}$$(49)
We need some of the following formulas to prove Proposition 2.3.
Lemma 2.4
For any \(\lambda \in {\mathbb{C}} {\setminus } \{0\},\) \(n \in {\mathbb{Z}},\) we have the following:
-
(1)
\(F_{\lambda }(n+1)=T_{\lambda }(n)F_{\lambda }(n),\)
-
(2)
\(\pi _{L}{\mathcal{C}}(n) T_{\lambda }(n-1)=\lambda \pi _{L},\)
-
(3)
\(\pi _{R} {\mathcal{C}}(n) T_{\lambda }(n)^{-1} =\lambda \pi _{R},\)
-
(4)
\({\mathrm{z}}_{L}(n)=\frac{\lambda }{a_{n+1}} T_{\lambda }(n)^{-1}\pi _{L} =\frac{\triangle _{n}}{d_{n}}{\mathcal{C}}(n)^{*}\pi _{L}=\frac{1}{\overline{a_{n}}}{\mathcal{C}}(n)^{*}\pi _{L},\)
-
(5)
\({\mathrm{z}}_{R}(n)=\frac{\lambda }{d_{n-1}} T_{\lambda }(n-1) \pi _{R} =\frac{\triangle _{n}}{a_{n}} {\mathcal{C}}(n)^{*} \pi _{R}=\frac{1}{\overline{d_{n}}} {\mathcal{C}}(n)^{*}\pi _{R},\)
-
(6)
\({\mathrm{z}}_{L}(n)^{*}+{\mathrm{z}}_{R}(n)={\mathrm{z}}_{L}(n)+{\mathrm{z}}_{R}(n)^{*}=I,\)
-
(7)
\(a_{n}{\mathrm{z}}_{L}(n)^{*}+d_{n}{\mathrm{z}}_{R}(n)^{*}=\pi _{L} {\mathcal{C}}(n) {\mathrm{z}}_{L}(n)^{*}+\pi _{R}{\mathcal{C}}(n){\mathrm{z}}_{R}(n)^{*}={\mathcal{C}}(n),\)
-
(8)
\(T_{\lambda }(n)[{\mathrm{z}}_{L}(n)-{\mathrm{z}}_{R}(n)]T_{1/\overline{\lambda }}(n)^{*}=[{\mathrm{z}}_{L}(n+1)-{\mathrm{z}}_{R}(n+1)].\)
Proof
(1) follows from (6), (2) and the first two equalities in (5) follow from (5), (11) and the unitarity of \({\mathcal{C}}(n).\) The inverse \(T_{\lambda }(n)^{-1}\) of the matrix \(T_{\lambda }(n)\) is given by
From this and the unitarity of \({\mathcal{C}}(n),\) (3) and the first two equalities in (4) follow. Since \({\mathcal{C}}(n)\) is unitary, we have
This and (11) show (6). By a direct computation using (11) and (51), we see
from which the item (7) follows. To prove (8), we first note that, by (51), the matrix \(T_{1/\overline{\lambda }}(n)^{*}\) is written as
From this and the item (4), we have
Using the concrete form for \(T_{1/\overline{\lambda }}(n)^{*}\) mentioned above, we also have
By a direct computation using the definition of \(T_{\lambda }(n)\) and the above formula for \({\mathrm{z}}_{R}(n)T_{1/\overline{\lambda }}(n)^{*},\) we have
Subtracting (54) from (53), we conclude (8). \(\square\)
Proof of Proposition 2.3
According to (11), the matrices \({\mathrm{z}}_{L}(n),\) \({\mathrm{z}}_{R}(n),\) \({\mathrm{z}}_{L}(n)-{\mathrm{z}}_{R}(n),\) and their adjoints are all nonzero for any \(n \in {\mathbb{Z}}.\) Since the matrix \(F_{\lambda }(n)\) is non-singular for any \(\lambda \in {\mathbb{C}} {\setminus } \{0\}\) and \(n \in {\mathbb{Z}},\) \(v_{\lambda }(n,m)\) and \(w_{\lambda }^{o}(n,m)\) are nonzero if and only if \(n \ne 0\) and m lies between 0 and n. In particular, the sums in (46) and (49) are finite. Thus, \(V_{\lambda }f\) and \(W_{\lambda }^{o}f\) are well defined as elements in \({\mathrm{Map}}\,({\mathbb{Z}},{\mathbb{C}}^{2})\) for any \(f \in {\mathrm{Map}}\,({\mathbb{Z}},{\mathbb{C}}^{2}).\)
First, we show the uniqueness of the solution to each of the initial value problems (27) and (28). Suppose that \(\Psi _{1}, \Psi _{2} \in {\mathrm{Map}}\,({\mathbb{Z}},{\mathbb{C}}^{2})\) solve the initial value problem (27). Then, we have \(\Psi _{1}(0)=\Psi _{2}(0)=w \in {\mathbb{C}}^{2}.\) We set \(\Psi =\Psi _{1}-\Psi _{2}.\) Then, \(\Psi\) satisfies \((U({\mathcal{C}})-\lambda )\Psi =0\) with \(\Psi (0)=0.\) According to Theorem 1.1, \(\Psi\) is in the image of the map \(\Phi _{\lambda } :{\mathbb{C}}^{2} \rightarrow {\mathrm{Map}}\,({\mathbb{Z}},{\mathbb{C}}^{2})\) defined in (7). Thus, we can write \(\Psi =\Phi _{\lambda }(u)\) with some \(u \in {\mathbb{C}}^{2}.\) Since \(F_{\lambda }(0)=I,\) we see \(0=\Psi (0)=\Phi _{\lambda }(u)(0)=F_{\lambda }(0)u=u,\) and hence, \(\Psi =\Phi _{\lambda }(0)=0.\) This shows that the solution to the initial value problem (27) is unique. Next, suppose that \(\Psi _{1},\Psi _{2}\) are solutions to the initial value problem (28). Then, we have \(\Psi _{1}(0)=\Psi _{2}(0)=w.\) We set \(\Psi =\Psi _{1}-\Psi _{2}.\) Then, \(\Psi\) satisfies \((U({\mathcal{C}})^{*}-\overline{\lambda }) \Psi =0\) with \(\Psi (0)=0.\) Applying \(U({\mathcal{C}})\) to the equation \(U({\mathcal{C}})^{*}\Psi =\overline{\lambda } \Psi\) and using Lemma 2.1, we have \(\Psi =\overline{\lambda } U({\mathcal{C}}) \Psi\) or, what is the same to say, \(U({\mathcal{C}})\Psi =\overline{\lambda }^{-1} \Psi\), because \(\lambda \ne 0\) as assumed. Therefore, by Theorem 1.1, \(\Psi\) is in the image of the map \(\Phi _{1/\overline{\lambda }} :{\mathbb{C}}^{2} \rightarrow {\mathrm{Map}}\,({\mathbb{Z}}, {\mathbb{C}}^{2}),\) and hence, we can write \(\Psi =\Phi _{1/\overline{\lambda }}(u)\) with some \(u \in {\mathbb{C}}^{2}.\) Then, we have \(0=\Psi (0)=\Phi _{1/\overline{\lambda }}(u)(0)=u\) by exactly the same discussion as before. Therefore, we see \(\Psi =\Phi _{1/\overline{\lambda }}(0)=0,\) which shows that the solution to (28) is unique.
Next, we check that the function defined by (45) (resp. (48)) solves the initial value problem (27) (resp. (28)). Since, by Theorem 1.1, we have \((U({\mathcal{C}})-\lambda ) \Phi _{\lambda }(w)=(U({\mathcal{C}})^{*}-\overline{\lambda })\Phi _{1/\overline{\lambda }}(w)=0\) and \(\Phi _{\lambda }(w)(0)=\Phi _{1/\overline{\lambda }}(w)(0)=w,\) it is enough to check that the function \(\Psi =V_{\lambda }f\) (resp. \(\Psi =W_{\lambda }^{o}f\)) solves the initial value problem (27) (resp. (28)) with the initial value \(w=0.\) The definitions (44) and (47) show that \(v_{\lambda }(0,m)=w_{\lambda }^{o}(0,m)=0\) for any \(m \in {\mathbb{Z}},\) and hence we see \(V_{\lambda }f(0)=W^{o}_{\lambda }f(0)=0.\) Thus, by Lemma 2.2, we need only to show that \(\Psi =V_{\lambda }f\) (resp. \(\Psi =W_{\lambda }^{o}f\)) satisfies the recurrence equation (31) (resp. (32)). Before proceeding to the proof that \(V_{\lambda }f\) satisfies (31), it is useful to give concrete formulas for \(T_{\lambda }v_{\lambda }(n,m).\) Namely, we have
These can be obtained directly from the definition (44) and the items (1), (4) in Lemma 2.4. (The item (4) in Lemma 2.4 is used only to obtain the formula when \(n=m \le -1.\)) In what follows, we use the notation, for example, ‘\(\overset{(1)}{=}\)’ to indicate that the item (1) in Lemma 2.4 is used to show the equality. We now show that \(\Psi =V_{\lambda }f\) satisfies (31). We have
which shows that \(V_{\lambda }f\) satisfies (31) when \(n=0\), because \(V_{\lambda }f(0)=0.\) To prove (31) with \(n=-1\) for \(\Psi =V_{\lambda }f,\) using (55), we see
which coincides with \(V_{\lambda }f(0)=0,\) and hence, \(V_{\lambda }f\) satisfies (31) when \(n=-1.\) Next, we consider the case \(n \ge 1.\) We have
which shows that the function \(V_{\lambda }f\) satisfies (31) for \(n \ge 1.\) For \(n \le -3,\) we calculate the right-hand side of (31) using (55) as follows:
For \(n=-2,\) also by (55), we see
This shows that \(V_{\lambda }f\) satisfies (31) for all \(n \in {\mathbb{Z}}.\) Next, to show that the function \(\Psi =W_{\lambda }^{o}f\) satisfies (32), we prepare the formulas for \(T_{1/\overline{\lambda }} w_{\lambda }^{o}(n,m)\) as follows:
For \(n=0,\) we have
which shows that \(W_{\lambda }^{o}f\) satisfies (32) for \(n=0.\) For \(n=-1,\) we calculate the right-hand side of (32) with \(n=-1,\) by (56), as
By the item (4) in Lemma 2.4, we have \({\mathrm{z}}_{L}(0)^{*}=\frac{1}{a_{0}}\pi _{L} {\mathcal{C}}(0).\) By the item (6) in Lemma 2.4, we have \(I-{\mathrm{z}}_{R}(-1)^{*}={\mathrm{z}}_{L}(-1).\) Substituting these formulas into (57), we obtain
This shows that \(W_{\lambda }^{o}f\) satisfies (32) for \(n=-1\), because \(W_{\lambda }^{o}f(0)=0.\) Let \(n \ge 1.\) We have
which shows that \(W_{\lambda }^{o}f\) satisfies (32) for \(n \ge 1.\) When \(n \le -2,\) we calculate the right-hand side of (32) for \(\Psi =W_{\lambda }^{o}f\) as
This shows that \(W_{\lambda }^{o}f\) satisfies (32) for any \(n \in {\mathbb{Z}}.\) \(\square\)
Corollary 2.5
For \(\lambda \in {\mathbb{C}}{\setminus } \{0\}\), we define \(w_{\lambda } \in {\mathrm{Map}}\,({\mathbb{Z}}^{2},{\mathrm{M}}_{2}({\mathbb{C}}))\) by
Let \(W_{\lambda } :C_{0}({\mathbb{Z}},{\mathbb{C}}^{2}) \rightarrow {\mathrm{Map}}\,({\mathbb{Z}},{\mathbb{C}}^{2})\) be a map defined by
Then, \(W_{\lambda }f\) satisfies
for any \(f \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2}).\)
Proof
We first note that \(W_{\lambda }\) defined in (59) is the formal adjoint operator of the linear map \(W_{\lambda }^{o} :C_{0}({\mathbb{Z}},{\mathbb{C}}^{2}) \rightarrow {\mathrm{Map}}\,({\mathbb{Z}},{\mathbb{C}}^{2})\).Footnote 3 We take \(f,g \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2}).\) Then, \((U({\mathcal{C}})-\lambda )f \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2}).\) By Proposition 2.3, (2) and taking \(w=0\) in (48), we have \((U({\mathcal{C}})^{*} -\overline{\lambda })W_{\lambda }^{o}g=g.\) Since \(W_{\lambda }\) is the formal adjoint operator of \(W_{\lambda }^{o} :C_{0}({\mathbb{Z}},{\mathbb{C}}^{2}) \rightarrow {\mathrm{Map}}\,({\mathbb{Z}},{\mathbb{C}}^{2}),\) we have
Since \(f,g \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2})\) is arbitrary, we have (60). \(\square\)
The function \(w_{\lambda }\) is given explicitly by the following:
One of the most important properties of the operator \(W_{\lambda }\) is the following.
Theorem 2.6
For any \(f \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2}),\) we have \(W_{\lambda }f \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2})\) and
where the QW-Fourier transform \(\widehat{f}^{{\mathcal{C}}}\) of f is defined in (16), and the function \(\delta _{m} \otimes u \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2})\) with \(m \in {\mathbb{Z}},\) \(u \in {\mathbb{C}}^{2}\) is defined in (9).
Proof
For fixed \(m \in {\mathbb{Z}},\) the function \(w_{\lambda }(n,m)\) in n can be nonzero only when n lies between 0 and m. Therefore, we have \(W_{\lambda }f \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2})\) for any \(f \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2}).\) Let \(f \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2})\) and \(n \ge 2.\) By (4) in Lemma 2.4 and (61), we have
By (1), (4), (5) in Lemma 2.4, we also have for \(m \ge n\)
and for \(m \ge n+2\)
Thus, the formula for \(m=n+1\) in (63) can be regarded as a special case of (64) and (65). We have
where we have used (1) in Lemma 2.4. Thus, \(U({\mathcal{C}}) W_{\lambda }f(n)\) can be calculated as follows:
We calculate this expression further. By Lemma 2.4, we see
for all \(n \in {\mathbb{Z}}.\) Using these formulas for \(n \ge 2,\) (66) can be further calculated as
which shows (62) for \(n \ge 2.\) Next, we consider the case \(n \le -2.\) By (5) in Lemma 2.4, we have
By (1), (4), (5) in Lemma 2.4, we also have for \(m \le n\)
and for \(m \le n-2\)
Thus, the second line in (68) is a special case of the formula for \(\pi _{R} {\mathcal{C}}(n-1) w_{\lambda } (n-1,m)\) in the case \(m \le n-2\) given above, and we have
Therefore, \(U({\mathcal{C}})W_{\lambda }f\) can be calculated as
which shows (62) for \(n \le -2.\) Since \(w_{\lambda }(0,0)=0,\) we have
For \(n=-1,\) we have
These calculations show that (62) holds also for \(n=\pm 1.\) Finally, we calculate \(U({\mathcal{C}})W_{\lambda }f (0).\) By Lemma 2.4 again, we see for \(m \ge 1\)
and for \(m \le -1\)
From this, we conclude
which shows that (62) holds for all \(n \in {\mathbb{Z}}.\) \(\square\)
Before proceeding to the proof of Theorem 1.2, we prove the Eq. (26) in Theorem 1.7 for \(f \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2}).\)
Proof of (26)
for \(f \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2}).\) The concrete form of the matrix \(T_{1/\overline{\lambda }}(n)^{*}\) is given in (52) and that of the matrix \(T_{1/\overline{\lambda }}(n)^{*-1}\) is given as
Using (52) and (70), it can be shown directly that
for all \(m \in {\mathbb{Z}}.\) From these formulas and (1) in Lemma 2.4, we have for \(f \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2})\)
which shows (26). \(\square\)
3 Green function and its properties
In this section, we investigate properties of the Green function (10) and give proofs of Theorems 1.2, 1.3 and 1.4. By the definition (10) of the Green function and (188), (189) in Appendix B, we see
It is well known that the resolvent \(R(\lambda )\) is holomorphic as a bounded operator-valued function in \(\lambda \in {\mathbb{C}} {\setminus } \sigma (U),\) and thus, \(R_{\lambda }(n,m)\) is holomorphic as an \({\mathrm{M}}_{2}({\mathbb{C}})\)-valued function in \(\lambda \in {\mathbb{C}} {\setminus } \sigma (U)\) for each fixed \(m,n \in {\mathbb{Z}}.\) For \(\lambda \in {\mathbb{C}} {\setminus } \sigma (U),\) we set
Using the concrete formula (30) of \(R(0)=U^{*},\) we have
for \(u \in {\mathbb{C}}^{2},\) and hence, \({\mathrm{x}}_{0}(0)=0.\) For any \(f \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2})\) and \(\lambda \in {\mathbb{C}} {\setminus } \sigma (U),\) the difference \(R(\lambda )f-V_{\lambda }f \in {\mathrm{Map}}\,({\mathbb{Z}},{\mathbb{C}}^{2}),\) where \(V_{\lambda }f\) is defined in (46), satisfies
by Proposition 2.3. Therefore, by Theorem 1.1, there exists a unique vector \({\mathrm{r}}_{\lambda }(f) \in {\mathbb{C}}^{2}\), such that
Since \(V_{\lambda }f(0)=0\) and \(\Phi _{\lambda }({\mathrm{r}}_{\lambda }(f))(0)={\mathrm{r}}_{\lambda }(f),\) we have
and for any \(u \in {\mathbb{C}}^{2}\)
More generally, we have
To prove Theorem 1.2, we need the following.
Lemma 3.1
For any \(n \in {\mathbb{Z}}\) and \(\lambda \in {\mathbb{C}} {\setminus } \{0\}\), we have
Proof
Since \(F_{\lambda }(0)=I\) for any \(\lambda \in {\mathbb{C}} {\setminus } \{0\},\) (76) for \(n=0\) is obvious. By the item (8) in Lemma 2.4, and the definition (6) of the matrix \(F_{\lambda }(n),\) we see
Let \(k \ge 1\) and suppose that (76) holds for \(n=k.\) By the items (1), (8) in Lemma 2.4, we see
hence, (76) holds for \(k+1.\) By induction, (76) holds for all \(n \ge 1.\) By the item (8) in Lemma 2.4, we see
and hence, we have
This shows that (76) holds for \(n=-1.\) Let \(k \le -1\) and suppose that (76) holds for \(n=k.\) Then, we have
which shows that (76) holds also for \(n=k-1.\) By induction, (76) holds for all \(n \le -1\) and, hence, for all \(n \in {\mathbb{Z}}.\) \(\square\)
Proof of Theorem 1.2
By the definition (46) of the operator \(V_{\lambda },\) we have
for any \(n,m \in {\mathbb{Z}}\) and \(u \in {\mathbb{C}}^{2}.\) Thus, by setting \(f=\delta _{m} \otimes u\) with \(m \in {\mathbb{Z}},\) \(u \in {\mathbb{C}}^{2}\) in (73), we have
Applying \(R(\lambda )\) to Eq. (62) in Theorem 2.6, we have
for \(f \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2})\) and \(n \in {\mathbb{Z}}.\) We set \(f=\delta _{m} \otimes u\) in the above. By the definition (16) of \({\mathcal{F}}_{{\mathcal{C}}},\) we have \({\mathcal{F}}_{{\mathcal{C}}}[\delta _{m} \otimes u](\lambda )=F_{1/\overline{\lambda }}(m)^{*}u.\) By the definition (59) of \(W_{\lambda },\) we see
Hence, substituting \(f=\delta _{m} \otimes u\) into (78), we obtain
Setting \(m=0\) in (77) gives
and substituting this into (79) yields
where we used the definition (44) of \(v_{\lambda }.\) For \(n=0,\) it follows from (80) and (61) that:
From (61), we have \(w_{\lambda }(n,m)=0\) for the two cases \((n \le -1,\, m \ge n+1)\) and \((n \ge 1,\, m \le n-1).\) Hence, we have
To consider the other cases, we note that by Lemma 3.1
Therefore, (80) gives
The formula (12) in Theorem 1.2 is a rewritten form of (81), (82), and (84). For \(n=m \ge 1,\) the formulas (80), (61) and Lemma 3.1 show
which implies (13) for \(n =m\ge 1.\) Similarly, for \(n \le -1,\) we have
which implies (13) for \(n=m \le -1\) and, hence, for all \(n \in {\mathbb{Z}}.\) \(\square\)
The Green function \(R_{\lambda }(n,m)\) is, as above, expressed in terms of the products \(F_{\lambda }(n)\) of the transfer matrices \(T_{\lambda }(n)\) and the special value \({\mathrm{x}}_{0}(\lambda )=R_{\lambda }(0,0)\) of the Green function. Therefore, we will face a computation of the matrix-valued function \({\mathrm{x}}_{0}(\lambda )\) when we apply results in this paper. Theorems 1.3 and 1.4 give us one of the methods to calculate \({\mathrm{x}}_{0}(\lambda )\) whose proof is given in the rest of this section. The following is one of the most important facts in the proof of Theorem 1.3.
Lemma 3.2
Let \(\lambda \in {\mathbb{C}} {\setminus } (\{0\} \cup S^{1}).\) Then, we have
In (85), the norm \(\Vert A\Vert _{{\mathrm{HS}}}\) of a \(2 \times 2\) matrix
is the Hilbert–Schmidt norm \(\Vert A\Vert _{{\mathrm{HS}}}\) defined by
Lemma 3.2 could be proved by a method in [17]. However, we give a proof different from it.
Proof
Let \(\lambda \in {\mathbb{C}} {\setminus } (\{0\} \cup S^{1}).\) First, we show the following formula:
Indeed, by (1), (2), (3) in Lemma 2.4, we have
which gives (86). To prove that
is infinity, we assume \(D<+\infty\) and deduce a contradiction. We note that \(D>0.\) In general, for \(C \in {\mathrm{U}}(2)\) and \(A, B \in {\mathrm{M}}_{2}({\mathbb{C}}),\) we have
Equation (86) and the identities in (87) imply
Therefore, we have
By definition, we have \(F_{\lambda }(0)=I\) and \(F_{\lambda }(1)=T_{\lambda }(0).\) These and the definition of \(T_{\lambda }(n)\) show
By substituting these into the above equation and by the assumption \(|\lambda | \ne 1,\) we conclude
which is a contradiction. Therefore, \(D=+\infty .\) Next, suppose that
We note that \(E>0.\) We have
Since \(F_{\lambda }(0)=I,\) we have \(\Vert \pi _{R}F_{\lambda }(0)\Vert _{{\mathrm{HS}}}^{2}=1.\) By the definition of \(F_{\lambda }(-1)=T_{\lambda }(-1)^{-1}\) and its concrete form (50), we see
Hence, we have
which is a contradiction. Therefore, we conclude \(E=+\infty .\) \(\square\)
Lemma 3.3
Let \(\lambda \in {\mathbb{C}} {\setminus } (\{0\} \cup S^{1}).\) Then, the dimensions of the subspaces \(A_{L}(\lambda )\) and \(A_{R}(\lambda )\) in Theorem 1.3are less than or equal to 1.
Proof
Suppose that \(\dim A_{L}(\lambda )=2.\) Then, we have
Summing these two sums gives \(\sum _{n \ge 1} \Vert F_{\lambda }(n)\Vert _{{\mathrm{HS}}}^{2}<+\infty ,\) which contradicts Lemma 3.2. Thus, we conclude \(\dim A_{L}(\lambda ) \le 1.\) Next, suppose that \(\dim A_{R}(\lambda )=2.\) Then, we see
Summing these two sums gives \(\sum _{n \le -1} \Vert F_{\lambda }(n)\Vert _{{\mathrm{HS}}}^{2}<+\infty ,\) which contradicts Lemma 3.2. Thus, we conclude \(\dim A_{R}(\lambda ) \le 1.\) \(\square\)
The function \(R(\lambda )(\delta _{0} \otimes u)\) is in \(\ell ^{2}({\mathbb{Z}},{\mathbb{C}}^{2})\) for any \(u \in {\mathbb{C}}^{2}\) and we have by Theorem 1.2
Since \(\dim A_{L} \le 1\) and \(\dim A_{R} \le 1\) by Lemma 3.3, we see
Proof of Theorem 1.3
It is left to prove that, for \(\lambda \in {\mathbb{C}} {\setminus } (\{0\} \cup S^{1}),\) the following holds:
To prove (89), suppose that \({\mathrm{rank}}[{\mathrm{x}}_{0}(\lambda ) +\lambda ^{-1} {\mathrm{z}}_{L}(0) ]=0.\) Then, \({\mathrm{x}}_{0}(\lambda )=-\lambda ^{-1} {\mathrm{z}}_{L}(0).\) By (88), we see
for any \(u \in {\mathbb{C}}^{2}.\) However, since \(\det ({\mathrm{z}}_{L}(0) -{\mathrm{z}}_{R}(0))= -\frac{\triangle _{0}}{a_{0}d_{0}},\) the matrix \({\mathrm{z}}_{L}(0) -{\mathrm{z}}_{R}(0)\) is non-singular. Thus, for any \(w \in {\mathbb{C}}^{2},\) there exists u, such that \(({\mathrm{z}}_{L}(0)-{\mathrm{z}}_{R}(0))u=w.\) Therefore, we have \(\sum _{n \le -1} \Vert F_{\lambda }(n)w\Vert _{{\mathbb{C}}^{2}}^{2}<+\infty\) for any \(w \in {\mathbb{C}}^{2}.\) This contradicts the fact that \(\dim A_{R}(\lambda )=1\) in Lemma 3.3. Hence, \({\mathrm{rank}}[{\mathrm{x}}_{0}(\lambda ) +\lambda ^{-1} {\mathrm{z}}_{L}(0) ]=1.\) Next, suppose that \({\mathrm{rank}}[{\mathrm{x}}_{0}(\lambda ) +\lambda ^{-1} {\mathrm{z}}_{R}(0) ]=0.\) Then, \({\mathrm{x}}_{0}(\lambda )=-\lambda ^{-1} {\mathrm{z}}_{R}(0).\) By (88), we see
for any \(u \in {\mathbb{C}}^{2}.\) As above, for any \(w \in {\mathbb{C}}^{2},\) there exists u, such that \(({\mathrm{z}}_{L}(0)-{\mathrm{z}}_{R}(0))u=w.\) Thus, we have \(\sum _{n \ge 1} \Vert F_{\lambda }(n)w\Vert _{{\mathbb{C}}^{2}}^{2}<+\infty\) for any \(w \in {\mathbb{C}}^{2}.\) This contradicts the fact that \(\dim A_{L}(\lambda )=1\) in Lemma 3.3. Hence, \({\mathrm{rank}}[{\mathrm{x}}_{0}(\lambda ) +\lambda ^{-1} {\mathrm{z}}_{R}(0) ]=1.\) \(\square\)
To prove Theorem 1.4, we use the following lemma.
Lemma 3.4
Let \(\lambda \in {\mathbb{C}} {\setminus } (\{0\} \cup S^{1})\) and \(u \in {\mathbb{C}}^{2}\) be arbitrary. Then, a vector \(w \in {\mathbb{C}}^{2}\) satisfies
if and only if \(w={\mathrm{x}}_{0}(\lambda )u.\)
Proof
Let \(\lambda \in {\mathbb{C}} {\setminus } (\{0\} \cup S^{1})\) and \(u,w \in {\mathbb{C}}^{2}.\) We define \(\Psi \in {\mathrm{Map}}\,({\mathbb{Z}},{\mathbb{C}}^{2})\) by
By Proposition 2.3, we have \(\Psi =V_{\lambda }(\delta _{0} \otimes u)+\Phi _{\lambda } (w).\) Therefore, again by Proposition 2.3, we have
Now, suppose that w satisfies (90). Then, \(\Psi\) defined above is in \(\ell ^{2}({\mathbb{Z}},{\mathbb{C}}^{2}).\) Therefore, by applying \(R(\lambda )\) to (92), we have
In particular, we have \(w=\Psi (0)=R_{\lambda }(0,0)u={\mathrm{x}}_{0}(\lambda )u\) by the definition (72) of \({\mathrm{x}}_{0}(\lambda ).\) Conversely, if \(w={\mathrm{x}}_{0}(\lambda )u,\) we have \(\Psi (n)=R_{\lambda }(n,0)u=R(\lambda )(\delta _{0} \otimes u)(n)\) by Theorem 1.2 which is in \(\ell ^{2}({\mathbb{Z}},{\mathbb{C}}^{2}).\) Hence, \(w={\mathrm{x}}_{0}(\lambda )u\) satisfies (90). \(\square\)
Proof of Theorem 1.4
Let \(\lambda \in {\mathbb{C}} {\setminus } (\{0\} \cup S^{1}).\) We first show that the unit vectors \({\mathrm{v}}_{+}(\lambda ),\) \({\mathrm{v}}_{-}(\lambda )\) form a basis in \({\mathbb{C}}^{2}.\) Indeed, suppose that \(a {\mathrm{v}}_{+}(\lambda )+b {\mathrm{v}}_{-}(\lambda )=0\) with \(a,b \in {\mathbb{C}}\) not simultaneously zero. If \(a =0\), then \(b{\mathrm{v}}_{-}(\lambda )=0\), and hence \(b=0\), because \({\mathrm{v}}_{-}(\lambda )\) is a nonzero vector in \({\mathbb{C}}^{2}.\) Thus, \(a \ne 0\), and thus, we can write \({\mathrm{v}}_{+}(\lambda )=c {\mathrm{v}}_{-}(\lambda )\) with \(c=-b/a.\) Then, by the property (14), the function \(\Phi _{\lambda }({\mathrm{v}}_{+}(\lambda ))\) is in \(\ell ^{2}({\mathbb{Z}},{\mathbb{C}}^{2})\), and hence, \(\lambda\) is in the spectrum \(\sigma (U),\) an eigenvalue, of the unitary operator \(U=U({\mathcal{C}})|_{\ell ^{2}({\mathbb{Z}},{\mathbb{C}}^{2})}.\) This is a contradiction, since we have assumed \(\lambda \not \in S^{1}.\) Thus, \({\mathrm{v}}_{\pm }(\lambda )\) forms a basis of \({\mathbb{C}}^{2}\), and hence, the matrix
is non-singular. We define \(a_{L}(\lambda ), b_{L}(\lambda ), a_{R}(\lambda ),b_{R}(\lambda ) \in {\mathbb{C}}\) by
Then, we claim that
From the definition (93) and that \(\Vert {\mathrm{v}}_{\pm }(\lambda )\Vert _{{\mathbb{C}}^{2}}=1,\) we have
Therefore, we have
Since \(\{{\mathrm{v}}_{+}(\lambda ), {\mathrm{v}}_{-}(\lambda )\}\) is a basis of \({\mathbb{C}}^{2},\) we obtain
Therefore, by (96), the vector \(w =b_{L}(\lambda ){\mathrm{v}}_{-}(\lambda )\) satisfies
since we have \({\mathrm{z}}_{R}(0)e_{L}=0\) and the vector \({\mathrm{v}}_{-}(\lambda )\) satisfies (14). By Lemma 3.4, we have \(b_{L}(\lambda ){\mathrm{v}}_{-}(\lambda )={\mathrm{x}}_{0}(\lambda )e_{L}.\) This and (96) show the first line of (94). Similarly, by (96), the vector \(w=a_{R}(\lambda ) {\mathrm{v}}_{+}(\lambda )\) satisfies
since we have \({\mathrm{z}}_{L}(0)e_{R}=0\) and the vector \({\mathrm{v}}_{+}(\lambda )\) satisfies (14). This and (96) show the second line of (94). For a unit vector \(u \in {\mathbb{C}}^{2},\) we take any unit vector \(u^{\perp }\) perpendicular to u. Since \(\{{\mathrm{v}}_{+}(\lambda ), {\mathrm{v}}_{+}(\lambda )^{\perp }\},\) \(\{{\mathrm{v}}_{-}(\lambda ), {\mathrm{v}}_{-}(\lambda )^{\perp }\}\) are orthonormal bases of \({\mathbb{C}}^{2},\) we see
and hence
Using this and (93), we have
Similarly, we have
Equations (94), (97) and (98) complete the proof of (15) in Theorem 1.4. \(\square\)
4 Integral representation for the Green function
As is seen in the previous sections, the function \({\mathrm{x}}_{0} :{\mathbb{C}} {\setminus } \sigma (U) \rightarrow {\mathrm{M}}_{2}({\mathbb{C}})\) introduced in (72) plays one of the central roles in the series of our results. However, since
it does not seem that the real part \([{\mathrm{x}}_{0}(\lambda ) +{\mathrm{x}}_{0}(\lambda )^{*}]/2\) has nice properties as the m-Carathéodory functions have. The definition of m-Carathéodory functions is given in the statement of Lemma 4.1 below. Instead of using \({\mathrm{x}}_{0},\) we use the following function:
By (99), we have
which shows the second equality in (23).
Lemma 4.1
The function \({\mathrm{x}} :{\mathbb{C}} {\setminus } \sigma (U) \rightarrow {\mathrm{M}}_{2}({\mathbb{C}})\) is an m-Carathéodory function in the sense that it is a holomorphic function on the unit disc with a positive real part and \({\mathrm{x}}(0)=I.\)
Proof
Since \({\mathrm{x}}_{0}\) is holomorphic on \({\mathbb{C}} {\setminus } \sigma (U),\) \({\mathrm{x}}\) is also holomorphic on \({\mathbb{C}} {\setminus } \sigma (U),\) and in particular, on the unit disc. By definition, we see \({\mathrm{x}}(0)=I.\) We set
For \(\lambda \in {\mathbb{C}} {\setminus } \sigma (U)\) and \(u ,v \in {\mathbb{C}}^{2},\) we have
and hence
This shows that
We note that
Therefore, the real part \({\mathrm{Re}}\,{\mathrm{x}}(\lambda )\) of \({\mathrm{x}}(\lambda )\) satisfies
if \(|\lambda |<1\) and \(u \ne 0.\) This completes the proof. \(\square\)
Therefore, there exists a unique positive \(2\times 2\)-matrix-valued measure
on \(S^{1},\) where \(\mu _{L}\) and \(\mu _{R}\) are probability measures and \(\alpha\) is a complex measure on \(S^{1},\) such that we have the following Herglotz representation:
We refer the readers to [8, 12] for the proof of the above fact. We denote C(X) the space of complex-valued continuous functions on a compact topological space X with the norm defined by \(\Vert f\Vert _{C(X)}=\sup _{x \in X}|f(x)|.\) The positive-matrix-valued measure \(\Sigma\) is characterized by
for any \(h \in C(S^{1})\) and \(u \in {\mathbb{C}}^{2},\) where \({\mathrm{d}}\ell (\zeta )\) denotes the Lebesgue measure on \(S^{1}\) with the unit total mass. In what follows, we choose the counterclockwise orientation on the unit circle \(S^{1}.\) Then, we can write
To prove Theorem 1.5, (1), we consider, for a fixed \(f \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2}),\) the following function:
The function \(H_{f}\) is holomorphic on the unit disc and, by (104), its real part is
Hence, \(H_{f}\) is a scalar Carathéodory function with \(H_{f}(0)=\Vert f\Vert ^{2}.\) Therefore, there exists a unique positive measure \(\nu _{f}\) on \(S^{1}\), such that
and the measure \(\nu _{f}\) is the unique measure satisfying
for any \(h \in C(S^{1}).\) In what follows, we prove the formula (17) with the matrix-valued measure appeared in (105) and (106). We need the following.
Lemma 4.2
The matrix-valued functions \(F_{\lambda }(n),\) \(F_{1/\overline{\lambda }}(n)^{*}\) are holomorphic in \(\lambda \in {\mathbb{C}} {\setminus } \{0\}\) for any fixed \(n \in {\mathbb{Z}}.\)
Proof
By (5) and (52), the matrix-valued functions \(T_{\lambda }(n),\) \(T_{1/\overline{\lambda }}(n)^{*}\) are non-singular and holomorphic in \(\lambda \in {\mathbb{C}} {\setminus } \{0\}\) for any \(n \in {\mathbb{Z}}.\) Therefore, their inverses are also holomorphic. By the definition (6), \(F_{\lambda }(n)\) is defined as a product of some \(T_{\lambda }(m)\)’s or their inverses. Thus, \(F_{\lambda }(n)\) is holomorphic in \(\lambda \in {\mathbb{C}} {\setminus } \{0\}.\) Similarly, \(F_{1/\overline{\lambda }}(n)^{*}\) is defined as a product of some \(T_{1/\overline{\lambda }}(m)^{*}\)’s or their inverses. Therefore, \(F_{1/\overline{\lambda }}(n)^{*}\) is also holomorphic. \(\square\)
Lemma 4.3
Let \(f \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2}).\) We set
Then, \(h_{f}(\lambda )\) is holomorphic on \({\mathbb{C}} {\setminus } \{0\}\), and we have \({\mathrm{Re}}\,h_{f}(\zeta )=\Vert \widehat{f}^{{\mathcal{C}}}(\zeta )\Vert _{{\mathbb{C}}^{2}}^{2}\) for \(\zeta \in S^{1}.\)
Proof
We first note that the statement on the analyticity of the function \(h_{f}(\lambda )\) is non-trivial, because it involves, at first glance, the matrix-valued function \({\mathrm{x}}_{0}(\lambda )\) which is holomorphic only on \({\mathbb{C}} {\setminus } \sigma (U).\) By the identity \(K(\lambda )=I+2\lambda R(\lambda )\) and the definition (16) of the QW-Fourier transform \(\widehat{f}^{{\mathcal{C}}},\) we see
The sums appeared in (112) are all finite sums, because f has a finite support. By Theorem 1.2, we have
Substituting these formulas into (112), we have
From (113) and Lemma 4.2, the function \(h_{f}(\lambda )\) is holomorphic in \(\lambda \in {\mathbb{C}} {\setminus }\{0\}.\) By taking the complex conjugate of (113), we see
By the item (6) in Lemma 2.4, we have \({\mathrm{z}}_{L}(0)^{*}=I-{\mathrm{z}}_{R}(0),\) \({\mathrm{z}}_{R}(0)^{*}=I-{\mathrm{z}}_{L}(0),\) and \({\mathrm{z}}_{L}(n)^{*}=I-{\mathrm{z}}_{R}(n).\) Substituting these identities into (114), we obtain
Gathering the first, third, and fifth lines in (115) and using (16), we obtain
In the second and the third lines in (116), we change the order of the summations in n and m, and exchange the role of the letters m, n; we see
In the last line of (117), we use Lemma 3.1, namely the formula
and we finally obtain
Now, we take \(\lambda =\zeta \in S^{1}.\) Since \(1/\overline{\zeta }=\zeta ,\) all of the terms in (113) and (118) are canceled each other after taking the sum of them except the term
in (118), and this shows the assertion. \(\square\)
We prepare an estimate of an integral which will be used to prove Proposition 4.5 below.
Lemma 4.4
Let \(u \in {\mathbb{C}}^{2}.\) Then, there exists a positive constant \(C_{u}\), such that, for any real number r satisfying \(2/3 \le r <1,\) we have
Proof
We take \(\zeta \in S^{1}\) and \(r \in [2/3,1).\) By definition, we have \({\mathrm{x}}(r\zeta )=I+2r\zeta {\mathrm{x}}_{0}(r\zeta ).\) This shows that
for any \(u \in {\mathbb{C}}^{2}.\) Taking \(f=\delta _{0} \otimes u\) in (109), we see
It follows from (119), (120) and the definition of \({\mathrm{x}}_{0}(\lambda )\) in (72) that:
Integrating this inequality over \(S^{1}\) with respect to the normalized Lebesgue measure \({\mathrm{d}}\ell ,\) and using the Cauchy–Schwarz inequality, we see
Now, from (111), we have
which shows that the integral \(\int _{S^{1}} {\mathrm{Re}}\,H_{\delta _{0} \otimes u}(r\zeta ) \,{\mathrm{d}}\ell (\zeta )\) is continuous in \(r \in [3/2,1].\) Hence, we can take a positive constant \(A_{u}\), such that
for any \(r \in [2/3,1].\) Therefore, we have
The assertion follows by setting \(C_{u}=\Vert u\Vert _{{\mathbb{C}}^{2}}/\sqrt{3}+3A_{u}.\) \(\square\)
The following proposition plays a central role in the proof of Theorem 1.5.
Proposition 4.5
For any \(f \in C_{0}({\mathbb{Z}},{\mathbb{C}}),\) the measure \(\nu _{f}\) on \(S^{1}\) given in (110), (111) is written as
where \({\mathrm{d}}\Sigma\) is the positive-matrix-valued measure given in (106), (107).
Proof
We use the characterization (111) of the measure \({\mathrm{d}}\nu _{f}.\) By Lemma 4.3, the function \(H_{f}(\lambda )\) can be written as
where \(h_{f}\) is defined in Lemma 4.3. Taking the real part, we see
By Lemmas 4.2 and 4.3, the functions \(\widehat{f}^{{\mathcal{C}}}(\lambda ),\) \(h_{f}(\lambda )\) are holomorphic on \({\mathbb{C}} {\setminus } \{0\}.\) Thus, by Lemma 4.3, we see
Therefore, (111) shows that
We calculate the right-hand side of (122) using (107) as follows. For simplicity of notation, we set
which are holomorphic on \({\mathbb{C}} {\setminus } \{0\}.\) We write
and hence
where we set
For \(0<r<1\) and \(\zeta \in S^{1},\) we set
so that
Since the functions \(F_{L},\) \(F_{R}\) are holomorphic on \({\mathbb{C}} {\setminus } \{0\},\) we have, for \(2/3 \le r <1\)
where we set
For simplicity, we denote \(k(r,\zeta )\) one of \(k_{L}(r,\zeta ),\) \(\overline{k_{L}(r,\zeta )},\) \(k_{R}(r,\zeta )\) and \(\overline{k_{R}(r,\zeta )},\) and we denote \(F(\lambda )\) one of \(F_{L}(\lambda ),\) \(\overline{F_{L}(\lambda )},\) \(F_{R}(\lambda )\) and \(\overline{F_{R}(\lambda )}.\) Then, for any \(h \in C(S^{1}),\) any \(u,v \in {\mathbb{C}}^{2},\) and any \(r \in [2/3, 1),\) Lemma 4.4 and the estimate (125) show
which tends to zero as \(r \uparrow 1.\) This shows that, for any \(h \in C(S^{1}),\) in the integral
the contributions from the first and the second term of \(g_{i}(r\zeta )\) in (124) tend to zero as \(r \uparrow 1.\) To handle the contributions from the last term involving \({\mathrm{Re}}\,{{\mathrm{x}}(r\zeta )}\) of \(g_{i}(r\zeta )\) in (124), let \(G(\lambda )\) be a smooth function on A and we take \(u,v \in {\mathbb{C}}^{2}.\) We write \(G(\lambda )=G(x,y)\) for \(\lambda =x+iy.\) Then, for \(\zeta \in S^{1},\) we have
where \(G_{x}\) (resp. \(G_{y}\)) is the partial derivative of G with respect to x (resp. y). Hence, for any \(h \in C(S^{1}),\) we see by Lemma 4.4
which tends to zero as \(r \uparrow 1.\) This and (107) show that
Therefore, by taking \(G(\lambda )\) as one of functions \(|F_{L}(\lambda )|^{2},\) \(F_{R}(\lambda )\overline{F_{L}(\lambda )},\) \(F_{L}(\lambda )\overline{F_{R}(\lambda )},\) \(|F_{R}(\lambda )|^{2},\) we obtain
Therefore, by (122), (123), we conclude
which completes the proof. \(\square\)
Proof of Theorem 1.5
The formula (121) in Proposition 4.5 is equivalent to the formula
Since \(H_{f}(\lambda )=\langle \,K(\lambda )f,f \,\rangle ,\) by setting \(\lambda =0\) in (127), we obtain
which shows (18) for \(f=g \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2}).\) Since \(K(\lambda )=I+2\lambda R(\lambda ),\) we see
which shows (17) for \(f=g \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2}).\) The polarization identity
shows (17) and (18) for any \(f,g \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2}).\) Hence, the assertion (2) in Theorem 1.5 has been proved. To prove the assertion (1) in Theorem 1.5, suppose that the positive-matrix-valued measure \(\Omega\) also satisfies the identity
for any \(f,g \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2}).\) We note that \(Uf \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2})\) for \(f \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2}).\) The formula (26) shows that for \(\lambda ,\mu \in {\mathbb{C}} {\setminus } \{0\}\)
Thus, replacing f by \((U-\lambda )f\) in (130) shows
Then, using the identity \(K(\lambda )=I+2\lambda R(\lambda )\) again, we have
From this, (103), and the identity \({\mathcal{F}}_{{\mathcal{C}}}[\delta _{0} \otimes u](\lambda )=u\) for any \(u \in {\mathbb{C}}^{2},\) we obtain the following Herglotz representation of the matrix \({\mathrm{x}}(\lambda ):\)
which shows \(\Sigma =\Omega .\) Thus, the uniqueness of the positive-matrix-valued measure in (17) and the assertion (1) in Theorem 1.5 have been proved. We remark that
for any \(h \in C(S^{1})\) and any \(f,g \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2}).\) Let us prove (131) before proceeding to the proof of the assertion (3) in Theorem 1.5. By the formula (26), we have
for any \(f \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2}).\) Iterated use of this and the formula (26) show that, for any integer n, positive or negative
By this formula and the item (2) in Theorem 1.5, we have
Therefore, for any Laurent polynomial p on \({\mathbb{C}} {\setminus } \{0\},\) we see
Since the set of Laurent polynomials in \(\zeta \in S^{1}\) is dense in \(C(S^{1})\) with respect to the norm \(\Vert \cdot \Vert _{C(S^{1})},\) and since \(\Vert h(U)\Vert _{{\mathrm{op}}} \le \Vert h\Vert _{C(S^{1})},\) where \(\Vert h(U)\Vert _{{\mathrm{op}}}\) denotes the operator norm of h(U) on \(\ell ^{2}({\mathbb{Z}},{\mathbb{C}}^{2}),\) we have (131). We now proceed to the proof of the assertion (3) in Theorem 1.5. For any \(h \in C(S^{1})\) and \(f \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2}),\) we have
Since the spectral measure \(\Vert E(\cdot )f\Vert ^{2}\) is the unique Borel measure on \(S^{1}\) satisfying the above with the total mass \(\Vert f\Vert ^{2},\) we see
for any Borel set A in \(S^{1},\) and hence, by the polarization identity (129), we see
Taking \(g=\delta _{n} \otimes v\) with \(n \in {\mathbb{Z}}\) and \(v \in {\mathbb{C}}^{2},\) and noting \({\mathcal{F}}_{{\mathcal{C}}}[\delta _{n} \otimes v](\lambda )=F_{1/\overline{\lambda }}(n)^{*}v,\) we have
Since (132) holds for any \(v \in {\mathbb{C}}^{2},\) we have (19). By setting \(A=S^{1}\) in (19), we conclude (20). \(\square\)
Proof of Corollary 1.6
By taking \(f=\delta _{0} \otimes u\) and \(n=0\) in (132), we have
We remark that the support, \({{\text{supp}}\,}(\Sigma ),\) of the positive-matrix-valued measure \(\Sigma\) on \(S^{1}\) is defined as the complement of the open set
Now, suppose that \(\zeta _{o} \in S^{1}\) is not contained in \(\sigma (U).\) Then, since \({{\text{supp}}\,}(E)=\sigma (U),\) there exists an open neighborhood U of \(\zeta _{o}\) in \(S^{1}\), such that \(E(U)=0.\) The identity (133) shows \(\Sigma (U)=0.\) Therefore, \(\zeta _{o} \not \in {{\text{supp}}\,}(\Sigma ).\) Conversely, suppose that \(\zeta _{o} \in S^{1}\) is not contained in \({{\text{supp}}\,}(\Sigma ),\) and let U be an open neighborhood of \(\zeta _{o}\) satisfying \(\Sigma (U)=0.\) This means that each of the entry of \(\Sigma\) is zero on U. Thus, the right-hand side of (132) with \(A=U\) is zero for any \(f \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2}),\) \(n \in {\mathbb{Z}}\) and \(v \in {\mathbb{C}}^{2}.\) This means that \(E(U)=0.\) Thus, we conclude that \(\zeta _{o} \not \in {{\text{supp}}\,}(E),\) which proves the assertion (1) in Corollary 1.6. Next, to prove the assertion (2) in Corollary 1.6, suppose that \(\lambda \in S^{1}\) satisfies \(\Sigma (\{\lambda \}) \ne 0.\) Then, by (133), we see \(\langle \,[E(\{\lambda \})(\delta _{0} \otimes u)](0),v \,\rangle _{{\mathbb{C}}^{2}} \ne 0\) for some \(u,u \in {\mathbb{C}}^{2}.\) Hence. \(E(\{\lambda \}) \ne 0\) and \(\lambda\) is an eigenvalue of U. Conversely, suppose that \(\lambda \in S^{1}\) is an eigenvalue of U. Let \(f \in \ell ^{2}({\mathbb{Z}},{\mathbb{C}}^{2})\) be an eigenfunction of U with the eigenvalue \(\lambda .\) Since \(E(\{\lambda \})\) is the projection onto the eigenspace of U with the eigenvalue \(\lambda ,\) we see \(E(\{\lambda \})f=f.\) By (132), we have
for any \(n \in {\mathbb{Z}}.\) The left-hand side of the above expression is nonzero for some \(n \in {\mathbb{Z}}\), because f is not identically zero. Therefore, \(\Sigma (\{\lambda \}) \ne 0,\) which proves the assertion (2) in Corollary 1.6. \(\square\)
Before proceeding to the proof of Theorem 1.7, we give, based on [7], some accounts on the sesquilinear form \(\langle \,\cdot ,\cdot \,\rangle _{\Sigma }\) defined in (24). In particular, we show that the positivity of the measure \(\langle \,{\mathrm{d}}\Sigma (\zeta )k(\zeta ),k(\zeta ) \,\rangle _{{\mathbb{C}}^{2}}\) for any \(k \in C(S^{1},{\mathbb{C}}^{2}).\) This fact can be proved using the characterization (107) of \(\Sigma\) and the non-negativity of \({\mathrm{Re}}\,{\mathrm{x}}(\lambda ).\) However, we give here an alternate proof.
We recall that, about the entries of positive-matrix-valued measure \(\Sigma ,\) the diagonals \(\mu _{L}\) and \(\mu _{R}\) are Borel probability measures, while \(\alpha =\langle \,\Sigma e_{L},e_{R} \,\rangle\) is a complex Borel measure satisfying \(|\alpha (A)|^{2} \le \mu _{L}(A) \mu _{R}(A)\) for any Borel set A, because \(\det \Sigma (A) \ge 0.\) From this, it follows that the total variation measure \(|\alpha |\) of \(\alpha\) also satisfies the inequality \(|\alpha |(A)^{2} \le \mu _{L}(A) \mu _{R}(A)\) for any Borel set A in \(S^{1}.\) This shows that \(\alpha ,\) \(|\alpha |\) is absolutely continuous with respect to both of \(\mu _{L}\) and \(\mu _{R}.\) We define a probability measure \(\mu\) by \(\mu (A)=(\mu _{L}(A)+\mu _{R}(A))/2,\) that is \(\mu ={\text{Tr}}(\Sigma )/2.\) Then, we have \(|\alpha |(A) \le \mu (A)\), and thus, \(\alpha ,\) \(\mu _{L}\) and \(\mu _{R}\) are all absolutely continuous with respect to the measure \(\mu ={\text{Tr}}(\Sigma )/2.\) We can write \(\alpha =\rho \mu ,\) \(\mu _{L}=\rho _{L} \mu ,\) \(\mu _{R}=\rho _{R} \mu\) with \(\rho _{L},\rho _{R},\rho \in L^{1}(S^{1},\mu ),\) where \(\rho _{L}\) and \(\rho _{R}\) are non-negative \(\mu\)-a.e. Then, we have \(|\alpha |=|\rho |\mu .\) For any \(\xi \in S^{1}\) and \(\varepsilon >0,\) we denote \(A(\xi ,\varepsilon )\) the arc in \(S^{1}\) centered at \(\xi\) and length \(\varepsilon .\) Then, by the Lebesgue differentiation theorem (see, for example, [4], 2.9.8 Theorem), we have
as \(\varepsilon \downarrow 0\) for \(\mu\)-a.e \(\xi \in S^{1}.\) Let X be a Borel set, such that \(\mu (X)=1\) and the above holds for any \(\xi \in X.\) Then, \(|\rho |^{2} \le \rho _{L} \rho _{R}\) on X. Let \(k \in C(S^{1},{\mathbb{C}}^{2})\) and we write \(k=\,\!^{t}(k_{L},k_{R})\) with \(k_{L},k_{R} \in C(S^{1}).\) Then, for any Borel set A on \(S^{1}\), we have
Since the integrand in (134) can be written as
it is non-negative at points where \(\rho _{L}>0\) in X. At a point \(\xi \in X\) satisfying \(\rho _{L}(\xi )=0,\) the integrand in (134) becomes \(|k_{R}(\xi )|^{2} \rho _{R}(\xi ) \ge 0.\) Thus, the integral in (134) is non-negative. Hence, the measure \(\langle \,{\mathrm{d}}\Sigma (\zeta )k(\zeta ), k(\zeta ) \,\rangle _{{\mathbb{C}}^{2}}\) is indeed a positive measure.
Proof of Theorem 1.7
For \(k \in C(S^{1},{\mathbb{C}}^{2}),\) we define \(\Vert k\Vert _{\infty }\) by the formula \(\Vert k\Vert _{\infty }=\sup _{\zeta \in S^{1}} \Vert k(\zeta )\Vert _{{\mathbb{C}}^{2}}.\) We first show the inequality
Indeed, using (134) for \(A=S^{1},\) we see
which shows (135). Let \(\pi\) be the natural projection from \(C(S^{1},{\mathbb{C}}^{2})\) to \(C(S^{1},{\mathbb{C}}^{2})/N.\) Then, the inner product \(\langle \,\cdot ,\cdot \,\rangle _{\Sigma }\) on \(C(S^{1},{\mathbb{C}}^{2})/N\) is defined as \(\langle \,\pi (k),\pi (l) \,\rangle _{\Sigma }=\langle \,k,l \,\rangle _{\Sigma }\) for \(k,l \in C(S^{1},{\mathbb{C}}^{2}),\) and \(L^{2}(S^{1},{\mathbb{C}}^{2})_{\Sigma }\) is the completion of \(C(S^{1},{\mathbb{C}}^{2})/N\) by the norm \(\Vert \pi (k)\Vert _{\Sigma }=\sqrt{\langle \,\pi (k),\pi (k) \,\rangle _{\Sigma }}.\) The inner product \(\langle \,\cdot , \cdot \,\rangle _{\Sigma }\) and its norm \(\Vert \cdot \Vert _{\Sigma }\) is defined on the whole of \(L^{2}(S^{1},{\mathbb{C}}^{2})_{\Sigma }\) in a standard manner. The map
is defined by the composition of \({\mathcal{F}}_{{\mathcal{C}}},\) \(\pi\) and the inclusion \(C(S^{1},{\mathbb{C}}^{2})/N \hookrightarrow L^{2}(S^{1},{\mathbb{C}}^{2})_{\Sigma }.\) For any \(u \in {\mathbb{C}}^{2}\) and \(n \in {\mathbb{Z}},\) we have, by (26), \({\mathcal{F}}_{{\mathcal{C}}}[U^{n} (\delta _{0} \otimes u)](\lambda )=\lambda ^{n}u.\) Therefore, the subspace \({\mathcal{F}}_{{\mathcal{C}}}[C_{0}({\mathbb{Z}},{\mathbb{C}}^{2})]\) of \(C(S^{1},{\mathbb{C}}^{2})\) contains the space of \({\mathbb{C}}^{2}\)-valued Laurent polynomials, and hence, it is dense in \(C(S^{1},{\mathbb{C}}^{2})\) with respect to the supremum norm \(\Vert \cdot \Vert _{\infty }.\) We take \(s \in L^{2}(S^{1},{\mathbb{C}}^{2})_{\Sigma }\) and \(\varepsilon >0.\) Then, we can take a function \(k \in C(S^{1},{\mathbb{C}}^{2})\), such that \(\Vert \pi (k) -s\Vert _{\Sigma }<\varepsilon .\) Since \({\mathcal{F}}_{{\mathcal{C}}}[C_{0}({\mathbb{Z}},{\mathbb{C}}^{2})]\) is dense in \(C(S^{1},{\mathbb{C}}^{2}),\) we can take a function \(f \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2})\), such that \(\Vert {\mathcal{F}}_{{\mathcal{C}}}(f) -k\Vert _{\infty }<\varepsilon .\) Therefore
This shows that \(\mathfrak {F}_{{\mathcal{C}}}[C_{0}({\mathbb{Z}},{\mathbb{C}}^{2})]\) is dense in \(L^{2}(S^{1},{\mathbb{C}}^{2})_{\Sigma }.\) For any \(f \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2}),\) (18) shows that \(\Vert \mathfrak {F}_{{\mathcal{C}}}(f)\Vert _{\Sigma }=\Vert f\Vert\) for \(f \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2}).\) To extend the map \(\mathfrak {F}_{{\mathcal{C}}}\) to the whole space \(\ell ^{2}({\mathbb{Z}},{\mathbb{C}}^{2}),\) we take \(f \in \ell ^{2}({\mathbb{Z}},{\mathbb{C}}^{2}).\) Noting that \(C_{0}({\mathbb{Z}},{\mathbb{C}}^{2})\) is dense in \(\ell ^{2}({\mathbb{Z}},{\mathbb{C}}^{2}),\) we take a sequence \(\{f_{n}\}_{n=1}^{\infty }\) of functions \(f_{n} \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2})\), such that \(\Vert f-f_{n}\Vert \rightarrow 0\) as \(n \rightarrow \infty .\) Then, we have \(\Vert \mathfrak {F}_{{\mathcal{C}}}(f_{n}) -\mathfrak {F}_{{\mathcal{C}}}(f_{m})\Vert _{\Sigma } =\Vert f_{n}-f_{m}\Vert \rightarrow 0\) as \(n,m \rightarrow 0,\) and thus, the limit \(s=\lim _{n \rightarrow \infty }\mathfrak {F}_{{\mathcal{C}}}(f_{n})\) exists in \(L^{2}(S^{1},{\mathbb{C}}^{2})_{\Sigma }.\) We have
We take another sequence \(\{g_{n}\}_{n=1}^{\infty }\) of functions \(g_{n} \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2})\) converging f. Then \(\Vert \mathfrak {F}_{{\mathcal{C}}}(g_{n}) -\mathfrak {F}_{{\mathcal{C}}}(f_{n})\Vert _{\Sigma }=\Vert g_{n}-f_{n}\Vert \rightarrow 0\) as \(n \rightarrow \infty .\) Hence
as \(n \rightarrow \infty .\) This shows that \(s=\lim _{n \rightarrow \infty }\mathfrak {F}_{{\mathcal{C}}}(f_{n})\) does not depend on the choice of sequences of functions in \(C_{0}({\mathbb{Z}},{\mathbb{C}}^{2})\) converging f. Thus, we can define \(\mathfrak {F}_{{\mathcal{C}}}(f)=\lim _{n \rightarrow \infty }\mathfrak {F}_{{\mathcal{C}}}(f_{n}).\) We prove that the map \(\mathfrak {F}_{{\mathcal{C}}} :\ell ^{2}({\mathbb{Z}},{\mathbb{C}}^{2}) \rightarrow L^{2}(S^{1},{\mathbb{C}}^{2})_{\Sigma }\) is unitary. That the map \(\mathfrak {F}_{{\mathcal{C}}}\) preserves the norm has proved in (137). To prove the surjectivity, we take \(s \in L^{2}(S^{1},{\mathbb{C}}^{2})_{\Sigma }.\) For each positive integer n, we take \(k_{n} \in C(S^{1},{\mathbb{C}}^{2})\) such that \(\Vert \pi (k_{n}) -s\Vert _{\Sigma }<1/n.\) Since \({\mathcal{F}}_{{\mathcal{C}}}[C_{0}({\mathbb{Z}},{\mathbb{C}}^{2})]\) is dense in \(C(S^{1},{\mathbb{C}}^{2}),\) we can take \(f_{n} \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2})\), such that \(\Vert {\mathcal{F}}_{{\mathcal{C}}}(f_{n}) -k_{n}\Vert _{\infty }<1/n.\) Then, we have
This shows that the sequence \(\{f_{n}\}\) converges to a function \(f \in \ell ^{2}({\mathbb{Z}},{\mathbb{C}}^{2})\) and \(\mathfrak {F}_{{\mathcal{C}}}(f)=\lim _{n \rightarrow \infty } \mathfrak {F}_{{\mathcal{C}}}(f_{n})=s.\) This shows that \(\mathfrak {F}_{{\mathcal{C}}}\) is surjective. Therefore, \(\mathfrak {F}_{{\mathcal{C}}} :\ell ^{2}({\mathbb{Z}},{\mathbb{C}}^{2}) \rightarrow L^{2}(S^{1},{\mathbb{C}}^{2})_{\Sigma }\) is a unitary operator. This completes the proof of Theorem 1.7. \(\square\)
5 Examples
In this section, we consider two examples, homogeneous quantum walks and certain quantum walks with non-constant coin matrix. The quantum walk with a non-constant coin matrix considered here is a special case of the so-called two phase models discussed originally in [3]. In these examples, we use Theorem 1.4 to compute concretely the matrix-valued function \({\mathrm{x}}_{0}(\lambda )\) and the positive-matrix-valued measure \(\Sigma .\)
5.1 Homogeneous quantum walks
First of all, let us consider the fundamental example, namely the case where the coin matrix \({\mathcal{C}}\) is constant, say \({\mathcal{C}}(n)=C \in {\mathrm{U}}(2)\) for any \(n \in {\mathbb{Z}}.\) For simplicity of notation, we assume
so that C can be written as
The transfer matrix \(T_{\lambda }(n)\) does not depend on \(n \in {\mathbb{Z}}\) and we denote it by \(T_{C}(\lambda ),\) which and whose inverse are given by
The matrix \(F_{\lambda }(n)\) is then given by
When \(\beta =0,\) we have \(|\alpha |=1\) and the transfer matrix \(T_{C}(\lambda )\) is a diagonal matrix, and the matrices \(F_{\lambda }(n),\) \(F_{1/\overline{\lambda }}(n)^{*}\) are given by
In this case, we can take
for the unit vectors \({\mathrm{v}}_{+}(\lambda ),\) \({\mathrm{v}}_{-}(\lambda )\) in Theorem 1.4. By the definition (11) of \({\mathrm{z}}_{L},\) \({\mathrm{z}}_{R},\) we have \({\mathrm{z}}_{L}(n)=\pi _{L},\) \({\mathrm{z}}_{R}(n)=\pi _{R}.\) By Theorem 1.4 and (21), we have
Therefore, using the characterization (107) of \(\Sigma ,\) it is concluded that the positive-matrix-valued measure \(\Sigma\) is given by \({\mathrm{d}}\Sigma (\zeta )=I {\mathrm{d}}\ell (\zeta ),\) the identity matrix times the normalized Lebesgue measure \({\mathrm{d}}\ell (\zeta ).\) The corresponding QW-Fourier transform \({\mathcal{F}}_{C}\) is given by
This is basically a usual Fourier series expansion. To be more precise, we define the Fourier series \({\mathcal{F}}[f]\) with \(f \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2})\) by
The function \({\mathcal{F}}[f]\) is a \({\mathbb{C}}^{2}\)-valued Laurent polynomial. We introduce a map \({\mathcal{J}}:C(S^{1},{\mathbb{C}}^{2}) \rightarrow C(S^{1},{\mathbb{C}}^{2})\) by
Then, we have
Next, we consider the case \(\beta \ne 0.\) In this case, we have \(0<|\alpha |<1.\) The characteristic equation for \(T_{C}(\lambda )\) is
Let S, T be subsets of \(S^{1}\) defined by
To calculate \(\Sigma ,\) we prepare the following lemma.
Lemma 5.1
The following holds.
-
(1)
The matrix \(T_{C}(\lambda )\) has an eigenvalue in \(S^{1}\) if and only if \(\lambda \in S.\)
-
(2)
\(T_{C}(\lambda )\) has an eigenvalue with multiplicity two if and only if \(\lambda\) is one of the following four points:
Proof
Suppose that \(T_{C}(\lambda )\) has an eigenvalue \(z_{o}\) satisfying \(|z_{o}|=1.\) Then, by (140), we have
This shows that \(J(\lambda )\) is real and \(|J(\lambda )| \le |\alpha |.\) The parameter \(\lambda\) satisfies the equation
Since r is real and \(|r| \le |\alpha |<1,\) we have \(\lambda \in S^{1}\) and \({\mathrm{Re}}\,(\lambda )=J(\lambda ).\) Therefore, \(|{\mathrm{Re}}\,(\lambda )| \le |\alpha |.\) Conversely, suppose that \(\lambda \in S^{1}\) and \(|{\mathrm{Re}}\,(\lambda )| \le |\alpha |.\) Since \(|\lambda |=1,\) we see \(J(\lambda )={\mathrm{Re}}\,(\lambda )\) and \(|J(\lambda )| \le |\alpha | <1.\) We write \(\lambda ={\mathrm{e}}^{i\theta }\) with \(\theta \in (0,\pi ).\) Then, \(\cos \theta =J(\lambda )\) and the solution to Eq. (140) is
Then, a direct calculation shows \(|z|=1,\) and which proves (1). Next, suppose that \(\lambda\) is one of the four points in (2). Then, \(J(\lambda )=\pm |\alpha |,\) and the discriminant of (140) is zero. Hence, \(T_{C}(\lambda )\) has an eigenvalue with multiplicity two. Conversely, suppose that \(T_{C}(\lambda )\) has an eigenvalue \(z_{o}\) with multiplicity two. Then, by (140), we have \(J(\lambda )^{2}=|\alpha |^{2}.\) Therefore, we see \(\lambda =J(\lambda ) \pm i \sqrt{1-|\alpha |^{2}}\) which coincides with one of the four points in the statement. \(\square\)
Thus, for \(\lambda \in {\mathbb{C}} {\setminus } (\{0\} \cup S^{1}),\) the matrix \(T_{C}(\lambda )\) has two mutually different eigenvalues, and, by (140), the absolute value of one of them is less than one and that of another is greater than one. We remark that, since \(T_{C}(\lambda )\) is holomorphic and its eigenvalues are simple for \(\lambda \in {\mathbb{C}} {\setminus } (\{0\} \cup S^{1}),\) the eigenvalues can be labelled, so that they are holomorphic there. Let \(z_{\pm }(\lambda )\) be the eigenvalues of \(T_{C}(\lambda )\) satisfying \(|z_{+}(\lambda )|<1<|z_{-}(\lambda )|\) for \(\lambda \in {\mathbb{C}} {\setminus } (\{0\} \cup S^{1})\) and holomorphic there.
Lemma 5.2
The function \(z_{+}(\lambda )\) is given by
where we denote \(\sqrt{z}\) for \(z \in {\mathbb{C}} {\setminus } (-\infty ,0]\) the branch of the square root whose values are positive for positive real numbers.
Proof
These expressions are solutions to Eq. (140). We also remark that the function \(\sqrt{J(\lambda )^{2}-|\alpha |^{2}}\) is holomorphic on \({\mathbb{C}} {\setminus } (S \cup i{\mathbb{R}})\) and the function \(\sqrt{|\alpha |^{2}-J(\lambda )^{2}}\) is holomorphic on \({\mathbb{C}} {\setminus } (T \cup {\mathbb{R}}).\) For a nonzero real number \(\lambda ,\) the function \(z_{+}(\lambda )\) is given by
which can be checked by the requirement \(|z_{+}(\lambda )|<1.\) The function in the first line of (144) is holomorphic on \(\{ \lambda \in {\mathbb{C}} {\setminus } \{0\} \mid {\mathrm{Re}}\,(\lambda )>0\} {\setminus } S,\) and that in the second line is holomorphic on \(\{ \lambda \in {\mathbb{C}} {\setminus } \{0\} \mid {\mathrm{Re}}\,(\lambda )<0\} {\setminus } S.\) Thus, these expressions of \(z_{+}(\lambda )\) still hold on the respective regions. We note that
A direct calculation shows
Thus, we see
and hence, we have
Other expressions of \(z_{+}(\lambda )\) are obtained in the same way. \(\square\)
Remark 5.3
As in the proof of Lemma 5.2, the function \(z_{+}(\lambda )\) is analytically continued through the relative interior of the subset T in \(S^{1}.\)
For the case of the constant coin, it is rather easy to use the usual Fourier series (139) to calculate the matrix \({\mathrm{x}}_{0}(\lambda ).\) Indeed, we have
where \(\widehat{U}(z)\) is the matrix-valued function given by
Calculating the integral (145) for \(n=m=0\) using the residue formula and the fact that \(z_{\pm }(\lambda )\) are the solutions to Eq. (140), we see
The formula (146) can also be obtained using Corollary 1.4, although it needs somehow complicated calculation. We give an outline of a calculation. We take the vectors \({\mathrm{v}}_{+}(\lambda ),\) \({\mathrm{v}}_{-}(\lambda )\) as eigenvectors corresponding to the eigenvalues \(z_{\pm }(\lambda ).\) Explicitly, we set
Suppose that \(0<\lambda <1.\) Then, \(\alpha z_{\pm }(\lambda )\) is real. Using the fact that \(z_{\pm }(\lambda )\) are the roots of (140), we find that the vectors \({\mathrm{v}}_{+}(\lambda )\) and \({\mathrm{v}}_{-}(\lambda )\) form an orthonormal basis in \({\mathbb{C}}^{2}\) for \(\lambda >0.\) By Theorem 1.4, we see
That the two formulas (148) and (146) are identical for \(0<\lambda <1\) can be verified by the formula
which hold for \(0<\lambda <1.\) Then, one can use the coincidence theorem for holomorphic function \({\mathrm{x}}_{0}(\lambda )\) to show that (146) holds also on the region \({\mathrm{Re}}\,(\lambda )>0,\) \(0<|\lambda |<1.\) The same discussion works well for other region in \({\mathbb{C}} {\setminus } (\{0\} \cup S^{1}).\) We note that from the formula (145), the Green function must satisfy
which is, according to (13) in Theorem 1.2, equivalent to
because \({\mathrm{z}}_{L}(n)={\mathrm{z}}_{L}(0).\) Equation (149) can be verified directly by (146) and the definition of \(T_{C}(\lambda ).\) A straightforward calculation shows
The positive-matrix-valued measure \(\Sigma\) is described as follows.
Theorem 5.4
The positive-matrix-valued measure \({\mathrm{d}}\Sigma (\zeta )\) is given by
where \(\chi _{\sigma (U)}\) is the characteristic function of the spectrum \(\sigma (U)=\{\zeta \in S^{1} \mid {\mathrm{Re}}\,\zeta \le |\alpha |\}\) of U.
Proof of Theorem 5.4
Let \(u \in {\mathbb{C}}^{2}\) and write it as
We need to calculate the limit
for \(h \in C(S^{1}).\) We have the expression
and hence
Thus, we need to calculate the limits of the integrals
as \(r \uparrow 1.\) We only consider the integral on the positive orthant \(\Delta =\{\zeta \in S^{1} \mid {\mathrm{Re}}\,(\zeta ) \ge 0, \mathrm{Im}\,(\zeta ) \ge 0\}\), because the other parts of the integral can be handled similarly. Thus, we set
Let \(\psi \in (0,\pi /2)\) be the real number satisfying \(\cos \psi =|\alpha |,\) and we divide the arc \(\Delta\) into two parts, \(D_{1}=\{0 \le t \le \psi \}\) and \(D_{2}=\{\psi \le t \le \pi /2\}.\) From Lemma 5.2, we have
By a direct calculation, we see
where we set
Since \(J(r) \ge 1,\) \(\sin (2t) \ne 0\) and \(D(\psi )=0,\) we can take a sufficiently small \(\varepsilon >0\), such that
Then, we see
for \(t \in [\psi ,\psi +\varepsilon ]\) and \(r \in [2/3,1].\) For \(t \in (\psi ,\pi /2],\) we have
Since \(1/D(t)^{1/2}\) is integrable on \([\psi ,\pi /2],\) the Lebesgue convergence theorem shows
On the interval \([\psi +\varepsilon ,\pi /2],\) D(t) is bounded, and thus, the Lebesgue convergence theorem is also applicable. Therefore, we see
On the arc \(D_{1},\) we can use the first expression of \(z_{+}(\lambda )\) in (143). We can still use the estimate (153) around \(t=\psi .\) However, this time, the imaginary part of \(z_{+}(r\zeta )/(z_{+}(r\zeta ) -z_{-}(r\zeta ))\) tends to zero as \(r \uparrow 1.\) Therefore, again, the Lebesgue convergence theorem shows
Thus, we obtain
The limit of the integral \(I_{r}\) also handled in the same way. This time, on the arc \(D_{1},\) the real part of \(K(\zeta )\) equals zero. On the arc \(D_{2},\) the real part of \(K(r\zeta )/\alpha (z_{+}(r\zeta ) -z_{-}(r\zeta ))\) converges, as \(r \uparrow 1,\) to \(\mathrm{Im}\,(\zeta ) /2\sqrt{|\alpha |^{2}-{\mathrm{Re}}\,(\zeta )^{2}}.\) Then, we obtain
The same calculations on other orthants combined with (152) show the assertion. \(\square\)
5.2 Simplest two-phase model
Let \(C_{0},\) \(C_{\pm }\) be three \(2 \times 2\) special unitary matrices, and we write them as
Let \({\mathcal{C}}:{\mathbb{Z}} \rightarrow {\mathrm{U}}(2)\) be a coin matrix defined as
The quantum walk \(U({\mathcal{C}})\) defined by the coin matrix \({\mathcal{C}}\) is called a two-phase model with one defect. Since the calculation is a bit complicated, we impose the following strong assumptions:
Therefore, \(C_{+}\) and \(C_{-}\) differs only by the phase of the diagonals. We note that the first two assumptions are for simplifying the calculation, but the last assumption is imposed for U to have eigenvalues. Eigenvalues of much general two-phase models are studied in [13]. In this case, the transfer matrix \(T_{\lambda }(n)\) satisfies
and the matrix-valued function \(F_{\lambda }(n)\) satisfies
As in the case of constant coins, we denote \(z_{\pm }(C_{\pm }, \lambda )\) the eigenvalues of \(T_{C_{\pm }}(\lambda )\) satisfying
and we define vectors \(w_{\pm }(C_{\pm }, \lambda )\) as
The vectors \(w_{\pm }(C_{\pm }, \lambda )\) are eigenvectors of \(T_{C_{\pm }}(\lambda )\) with the eigenvalues \(z_{\pm }(C_{\pm }, \lambda ),\) respectively. We define unit vectors \({\mathrm{v}}_{\pm }(\lambda )\) by
These vectors satisfy
From (161), it follows that we can use the unit vectors \({\mathrm{v}}_{+}(\lambda ),\) \({\mathrm{v}}_{-}(\lambda )\) to apply Theorem 1.4. We set
Then, by Lemma 5.1, for \(\lambda \in {\mathbb{C}} {\setminus } \{0\},\) we have \(|z_{\pm }(C_{\pm },\lambda )|=1\) if and only if \(\lambda \in S.\)
Lemma 5.5
There are no eigenvalues of U in S.
Proof
This is proved in [17], but we give a proof for completeness. Suppose contrary that U has an eigenvalue \(\lambda\) in S. Then, there exists a nonzero vector \(u \in {\mathbb{C}}^{2}\), such that the function \(\Phi _{\lambda }(u)\) defined in (7) is in \(\ell ^{2}({\mathbb{Z}},{\mathbb{C}}^{2}).\) Since \(\lambda \in S,\) all the eigenvalues of \(T_{C_{\pm }}(\lambda )\) are in \(S^{1}\) by Lemma 5.1. By an argument using the Jordan normal form of \(T_{C_{\pm }}(\lambda ),\) we can find a positive constant c, such that
for any \(w \in {\mathbb{C}}^{2}\) and any positive integer n. By (157), we have
Since \(\Phi _{\lambda }(u) \in \ell ^{2}({\mathbb{Z}},{\mathbb{C}}^{2}),\) we must have \(T_{\lambda }(0)u=T_{\lambda }(-1)^{-1}u=0.\) This implies that \(u=0\) which is a contradiction. \(\square\)
The matrices \(T_{\lambda }(0),\) \(T_{\lambda }(-1)\) and their inverses are given by
Then, the concrete form of the vectors \({\mathrm{v}}_{\pm }(\lambda )\) is given by
where \(D_{\pm }(\lambda )\) are normalization terms and \(Z_{\pm }(\lambda )\) are given by
By the assumption (156), \(z=Z_{\pm }(\lambda )\) are the solutions to the equation
In the case of constant coins, the denominators \(\langle \,{\mathrm{v}}_{+}(\lambda ), {\mathrm{v}}_{-}(\lambda )^{\perp } \,\rangle _{{\mathbb{C}}^{2}}\) and \(\langle \,{\mathrm{v}}_{-}(\lambda ), {\mathrm{v}}_{+}(\lambda )^{\perp } \,\rangle _{{\mathbb{C}}^{2}}\) in (15) are constant functions in \(\lambda\) and they do not contribute the asymptotic behavior of \({\mathrm{x}}_{0}(\lambda )\) as \(|\lambda | \rightarrow 1.\) However, in the two-phase model, there are points \(\zeta \in S^{1}\) where these denominators are asymptotically zero as \(\lambda \rightarrow \zeta \in S^{1}.\) We need to describe these points and analyze the behavior of \({\mathrm{x}}_{0}(\lambda )\) near these points. We set
In what follows, we use the unit vector \(u^{\perp }\) perpendicular to a unit vector u defined by
Lemma 5.6
The denominators are given by
Proof
Since \(Z_{\pm }(\lambda )\) are the solution to Eq. (164), they satisfy \(Z_{+}(\lambda )-\lambda ^{-1}=Z_{-}(\lambda )-\lambda\) and \(\rho ^{2}=2J(\lambda )Z_{-}(\lambda )-Z_{-}(\lambda )^{2}.\) Thus, we see
Therefore, a direct calculation using the property \(|\alpha _{0}|^{2}+|\beta _{0}|^{2}=1\) shows
Substituting \(2J(\lambda )Z_{-}(\lambda )-Z_{-}(\lambda )^{2}=\rho ^{2}=1-\beta \overline{\beta }\) into the above, we see
By the choice of the vector \({\mathrm{v}}_{\pm }(\lambda )^{\perp }\) perpendicular to \({\mathrm{v}}_{\pm }(\lambda )\) described in (166), we have
which completes the proof. \(\square\)
In what follows, we sometimes write
for a function G on \({\mathbb{C}} {\setminus } S^{1},\) if the limit exists. We write \(\zeta =z+iy \in S^{1}.\) We define a function \(f(\lambda )\) by
Then, by Lemma 5.2, for \(\zeta \not \in S,\) we have
where
Thus, \(f(\zeta )=0\) if and only if \(\zeta\) is one of the following four points:
where positive real numbers \(x_{*},\) \(y_{*}\) are given by
It is well known (see [13]) that these points are actually the eigenvalues of U, which will also be shown as point masses of \(\Sigma\) in the following theorem.
Theorem 5.7
The positive-matrix-valued measure \(\Sigma\) for the two-phase model \(U({\mathcal{C}})\) with the assumption (156) is given by the following:
where we write \(\zeta =x+iy \in S^{1}\) with \(x,y \in {\mathbb{R}},\) and \(\zeta _{*}=x_{*}+iy_{*}\) is defined in (170).
Proof of Theorem 5.7
By (167), we have
Therefore, by Theorem 1.4, we have
where the holomorphic function \(f(\lambda )\) is given in (168). We first calculate the matrix-valued measure \(\Sigma\) on S. We only consider the upper part \(S_{+}=\{\zeta \in S \mid \mathrm{Im}\,(\zeta )>0\}\) of S. The measure \(\Sigma\) on the lower part of S can be calculated in the same way. For \(\zeta \in S_{+},\) by Lemma 5.2, we have
Therefore, writing \(\zeta =x+iy\) as before
By the assumption (156), we have \(|s|<1-\rho ^{2}\), and hence, \(f(\zeta _{(\pm )})\) are nonzero. Since both of \(\pm \sqrt{\rho ^{2}-J(\lambda )^{2}}\) are continuous on S, the Lebesgue convergence theorem shows
Next, we calculate the matrix-valued measure \(\Sigma\) on \(S^{1} {\setminus } S.\) We have
Therefore, we only need to calculate the behavior of \({\mathrm{x}}_{0}(\lambda )\) near \(\{\pm \zeta _{*}, \pm \overline{\zeta _{*}} \}.\) On the domains
the function \(Z_{-}(\lambda )\) is given as
We still use the notation \(f(\lambda )=1-s-J(\lambda )Z_{-}(\lambda )\) on the domains \(\Omega _{\pm }.\) We denote \(\zeta _{o}=x_{o}+iy_{o}\) one of the points \(\{\pm \zeta _{*},\pm \overline{\zeta _{*}}\}\) and write \({\mathrm{e}}^{i\theta _{o}}=\zeta _{o}.\) For any small \(\varepsilon >0\) and \(r \in (2/3,1)\) close to 1, we define
where \(\varepsilon >0\) is chosen, so that \(J_{r,\varepsilon }\) is contained in one of \(\Omega _{\pm }\) containing \(\zeta _{o}\) and \(J_{r,\varepsilon }\) contains only \(\zeta _{o}\) among \(\{\pm \zeta _{*},\pm \overline{\zeta _{*}}\}.\) We note that, by Lemma 5.6, \(f(\lambda )\) is nonzero for \(|\lambda | \ne 1\) since, for \(|\lambda | \ne 1,\) \(\{{\mathrm{v}}_{+}(\lambda ),\, {\mathrm{v}}_{-}(\lambda )\}\) is a basis of \({\mathbb{C}}^{2}.\) Then, \(\zeta _{o}\) is a zero of \(f(\lambda )\) of order 1 and there are no zeros of \(f(\lambda )\) on a neighborhood of \(J_{r,\varepsilon },\) where the function \(f(\lambda )\) is defined in (168) which is holomorphic on \(\Omega _{\pm }.\) Thus, for any holomorphic function \(h(\lambda )\) near \(J_{r,\varepsilon },\) Cauchy’s integral formula gives
where \(\partial J_{r,\varepsilon }\) is the boundary of \(J_{r,\varepsilon }\) with the counterclockwise direction. Since \(Z_{-}(\lambda )\) satisfies the Eq. (164), we have
Thus, we have \(f'(\lambda )=-Z_{-}(\lambda )Z_{-}'(\lambda ).\) Substituting this into (174) yields
By calculating the contour integral in the left-hand side of (175), we see
The set of all Laurent polynomials is dense in the space of continuous functions on \(A_{\varepsilon }.\) Thus, the formula (176) still holds for any continuous function h on \(A_{\varepsilon }.\) Let \(u=\,\!^{t}[u_{L}, u_{R}] \in {\mathbb{C}}^{2}\) and \(h \in C(A_{\varepsilon }).\) According to the formula (171), we set
so that
A direct calculation shows
where, as before, \(K(\lambda )=(\lambda -\lambda ^{-1})/2.\) Since \(0=f(\zeta _{o})=1-s-x_{o}Z_{-}(\zeta _{o}),\) when \(x_{o}={\mathrm{Re}}\,(\zeta _{o})>0\) (that is, \(x_{o}=x_{*}\)), we have
Then, we also have
Therefore, from (177), we obtain, when \(x_{o}={\mathrm{Re}}\,(\zeta _{o})>0\)
where \(\delta _{\zeta _{o}}\) is the delta measure at the point \(\zeta _{o},\) and for \({\mathrm{Re}}\,(\zeta _{o})<0\)
From this, we conclude the assertion in Theorem 5.7. \(\square\)
Notes
See Appendix B for the notion of formal adjoint operators and their construction.
References
Cantero, M.J., Grünbaum, F.A., Moral, L., Velázquez, L.: Matrix-valued Szegö polynomials and quantum random walks. Commun. Pure Appl. Math. 63(4), 464–507 (2010)
Cantero, M.J., Moral, L., Velázquez, L.: Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle. Linear Algebra Appl. 362, 29–56 (2003)
Endo, T., Konno, N., Segawa, E., Takei, M.: Limit theorems of a two-phase quantum walk with one defect. Quantum Inf. Comput. 15, 1373–1396 (2015)
Federer, H.: Geometric Measure Theory. Classics in Mathematics (Reprint of the 1969 Ed.). Springer, Berlin (1996)
Fillman, J., Ong, D.C.: Purely singular continuous spectrum for limit-periodic CMV operators with applications to quantum walks. J. Funct. Anal. 272(12), 5107–5143 (2017)
Grimmett, G., Janson, S., Scudo, P.: Weak limits for quantum random walks. Phys. Rev. E 69, 026119 (2004)
Gesztesy, F., Tsekanovskii, E.: On matrix-valued Herglotz functions. Math. Nachr. 218(1), 61–138 (2000)
Gesztesy, F., Zinchenko, M.: Weyl–Titchmarsh theory for CMV operators associated with orthogonal polynomials on the unit circle. J. Approx. Theory 139(1–2), 172–213 (2006)
Konno, N.: A new type of limit theorems for the one-dimensional quantum random walk. J. Math. Soc. Jpn. 57, 1179–1195 (2005)
Kodaira, K.: The eigenvalue problem for ordinary differential equations of the second order and Heisenberg’s theory of S-matrices. Am. J. Math. 71(4), 921–945 (1949)
Kawai, H., Komatsu, T., Konno, N.: Stationary measure for two-state space-inhomogeneous quantum walk in one dimension. Yokohama Math. J. 63, 59–74 (2017)
Kotani, K., Matano, H.: Differential Equations and Eigenfunction Expansion. Iwanami Shoten, Tokyo (2006) (in Japanese)
Kiumi, C., Saito, K.: Eigenvalues of two-phase quantum walks with one defect in one dimension. Quantum Inf. Process. 20, 11 (2021) (Article no. 171)
Marchenko, V.A.: Sturm–Liouville Operators and Applications. Birkhäuser, Basel (1986)
Morioka, H.: Generalized eigenfunctions and scattering matrices for position-dependent quantum walks. Rev. Math. Phys. 31(7), 1950019 (2019)
Morioka, H., Segawa, E.: Detection of edge defects by embedded eigenvalues of quantum walks. Quantum Inf. Process. 18(2), 18 (2019) (Article no. 283)
Maeda, M., Sasaki, H., Segawa, S., Suzuki, S., Suzuki, K.: Dispersive estimates for quantum walks on 1D lattice. J. Math. Soc. Japan 74(1), 217–246 (2022)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics, II: Fourier Analysis, Self-Adjointness. Academic Press, Inc., San Diego (1975)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics, III: Scattering Theory. Academic Press, Inc., San Diego (1979)
Richard, A., Suzuki, R., de Aldecoa, T.: Quantum walks with an anisotropic coin, I: spectral theory. Lett. Math. Phys. 108, 331–357 (2018)
Richard, A., Suzuki, R., de Aldecoa, T.: Quantum walks with an anisotropic coin, II: scattering theory. Lett. Math. Phys. 109, 61–88 (2019)
Simon, B.: Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory, Colloquium Publications, Part 1, vol. 54. American Mathematical Society, Providence (2005)
Simon, B.: Orthogonal Polynomials on the Unit Circle, Part 2: Spectral Theory. American Mathematical Society Colloquium Publications, Part 2, vol. 54. American Mathematical Society, Providence (2005)
Stone, M.H.: Linear Transformations in Hilbert Space. American Mathematical Society Colloquium Publications, vol. 15. American Mathematical Society, Providence (1932)
Sunada, T., Tate, T.: Asymptotic behavior of quantum walks on the line. J. Funct. Anal. 262(6), 2608–2645 (2012)
Titchmarsh, E.C.: Eigenfunction Expansion, Part I. Oxford University Press, Oxford (1962)
Weyl, H.: Über gewöhnliche Differentialgleichungen mit singulären Stellen und ihre Eigenfunktionen. Göttinger Nachrichten 230–254 (1935)
Weyl, H.: Über gewöhnliche Differentialgleichungen mit singulären Stellen und ihre Eigenfunktionen. Göttinger Nachrichten 442–467 (1910)
Acknowledgements
The author would like to thank the anonymous referees for their comments and suggestions that help improve the presentation of the paper. The author is partially supported by JSPS KAKENHI Grant No. 18K03267, 17H06465.
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Communicated by George Androulakis.
Appendices
Appendix A: Proof of Theorem 1.1
Theorem 1.1 is proved in [11, 13, 17]. However, the way of presentation here is somehow different from them. Thus, we give its proof for completeness. Suppose that \(f \in {\mathrm{Map}}\,({\mathbb{Z}},{\mathbb{C}}^{2})\) satisfies \(U({\mathcal{C}})f=\lambda f\) with \(\lambda \in {\mathbb{C}} {\setminus } \{0\}.\) We write
Then, by the definition of \(U({\mathcal{C}}),\) we have
Shifting the variable n, we get
The second equation of (181) gives
Substituting this into the first equation of (180) shows
Equations (182) and (183) give
where \(T_{\lambda }(n)\) is defined in (5). From (184) and the definition (6) of the matrix \(F_{\lambda }(n),\) we have \(f(n)=F_{\lambda }(n)f(0)=\Phi _{\lambda }(f(0))(n)\), and hence, \(f \in \Phi _{\lambda }({\mathbb{C}}^{2}).\) Conversely, we consider the function \(f(n):=\Phi _{\lambda }(u)(n)\) defined in (7) with a nonzero vector \(u \in {\mathbb{C}}^{2}.\) Using Lemma 2.4, we see
which shows that \(f \in {\mathcal{M}}^{\lambda }.\) Therefore, we have \({\mathcal{M}}^{\lambda }=\Phi _{\lambda }({\mathbb{C}}^{2}).\) Since \(F_{\lambda }(n)\) is an invertible matrix for each \(\lambda \in {\mathbb{C}} {\setminus } \{0\}\) and \(n \in {\mathbb{Z}},\) the map \(\Phi _{\lambda } :{\mathbb{C}}^{2} \rightarrow {\mathrm{Map}}\,({\mathbb{Z}},{\mathbb{C}}^{2})\) is injective. This shows that \(\dim {\mathcal{M}}^{\lambda }=\dim \mathrm{Im}\,\Phi _{\lambda }=2.\) Next, we introduce a map
defined as
Let \(f=\Phi _{\lambda }(w)\) with \(w \in {\mathbb{C}}^{2}.\) Using (181) we see
This is equivalent to
for all \(w \in {\mathbb{C}}^{2}.\) The matrix \(S_{\lambda }(n)\) is also called the transfer matrix [13, 17]. Compared with \(T_{\lambda }(n)\) which we used throughout the paper, \(S_{\lambda }(n)\) has a nice property; \(\det S_{\lambda }(n)=d_{n}/a_{n}\), and hence, \(|\det S_{\lambda }(n)|=1\), whereas \(\det T_{\lambda }(n) =d_{n}/a_{n+1}\) whose absolute value is, in general, not equal to 1. We take \(v,w \in {\mathbb{C}}^{2}\) and suppose that \(\Phi _{\lambda }(v) \in \ell ^{2}({\mathbb{Z}},{\mathbb{C}}^{2})\) and \(\Vert \Phi _{\lambda }(w)(n)\Vert _{{\mathbb{C}}^{2}}\) is bounded. By the definition of J, \(\Vert (J\Phi _{\lambda }(w))(n)\Vert _{{\mathbb{C}}^{2}}\) is bounded. Since J is unitary, \(J\Phi _{\lambda }(v) \in \ell ^{2}({\mathbb{Z}},{\mathbb{C}}).\) Thus, we have \(\Vert (J\Phi _{\lambda }(v))(n)\Vert _{{\mathbb{C}}^{2}} \rightarrow 0\) as \(n \rightarrow \infty .\) We define a function \(W :{\mathbb{Z}} \rightarrow {\mathbb{R}}\) by the formula
By (185), we see that
for each \(n \in {\mathbb{Z}}.\) Therefore, W is a constant function. Since \(\Vert (J\Phi _{\lambda }(v))(n)\Vert _{{\mathbb{C}}^{2}} \rightarrow 0,\) we have \(W(n) \rightarrow 0\) as \(n \rightarrow \infty .\) Hence, W(n) is identically zero. This shows that \(J\Phi _{\lambda }(v)(n)\) and \(J\Phi _{\lambda }(w)(n)\) are linearly dependent for each \(n \in {\mathbb{Z}}.\) We take \(s_{0}, t_{0} \in {\mathbb{C}}\) which is not simultaneously zero satisfying
Then, by (185), we have, when \(n \ge 1\)
Therefore, we have \(s_{0} J\Phi _{\lambda }(v) +t_{0}J\Phi _{\lambda }(w)=0\) on \({\mathbb{Z}}\), and this shows \(\dim {\mathcal{M}}^{\lambda } \cap \ell ^{2}({\mathbb{Z}},{\mathbb{C}}^{2}) \le 1.\) \(\square\)
Appendix B: Formal adjoint operators
In this section, we prove the following.
Lemma B.1
Let \(A :C_{0}({\mathbb{Z}},{\mathbb{C}}^{2}) \rightarrow {\mathrm{Map}}\,({\mathbb{Z}},{\mathbb{C}}^{2})\) be a linear map. Then, there exists a unique linear operator \(A^{*} :C_{0}({\mathbb{Z}},{\mathbb{C}}^{2}) \rightarrow {\mathrm{Map}}\,({\mathbb{Z}},{\mathbb{C}}^{2})\) satisfying
We call the operator \(A^{*}\) the formal adjoint of A.
Before proceeding to the proof of Lemma B.1, let us prepare some notation and simple properties of a linear map \(A :C_{0}({\mathbb{Z}},{\mathbb{C}}^{2}) \rightarrow {\mathrm{Map}}\,({\mathbb{Z}},{\mathbb{C}}^{2}).\) We recall that, for \(m \in {\mathbb{Z}}\) and \(u \in {\mathbb{C}}^{2},\) the function \(\delta _{m} \otimes u \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2})\) is defined as in (9). Then, any function \(g \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2})\) can be written as
Let \(A :C_{0}({\mathbb{Z}}, {\mathbb{C}}^{2}) \rightarrow {\mathrm{Map}}\,({\mathbb{Z}},{\mathbb{C}}^{2})\) be a linear map. For any \(m,n \in {\mathbb{Z}},\) we define a linear map \(k_{A}(n,m) :{\mathbb{C}}^{2} \rightarrow {\mathbb{C}}^{2}\) by
The \(2 \times 2\) matrix-valued function \(k_{A}(m,n)\) is called the kernel function of A. The sum in (187) is finite for \(g \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2}),\) and hence, we have
Proof of Lemma B.1
First, we show that the operator satisfying (186) is unique. Indeed, suppose that there are two linear maps B, C from \(C_{0}({\mathbb{Z}},{\mathbb{C}}^{2})\) to \({\mathrm{Map}}\,({\mathbb{Z}},{\mathbb{C}}^{2})\) satisfying
We set \(X=B-C.\) Then, we have \(\langle \,Xf,g \,\rangle =0\) for any \(f,g \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2}).\) For any \(n \in {\mathbb{Z}},\) \(u \in {\mathbb{C}}^{2}\) and \(f \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2}),\) we see
This shows that \(\langle \,(Xf)(n),u \,\rangle _{{\mathbb{C}}^{2}}=0\) for any \(u \in {\mathbb{C}}^{2},\) and hence \((Xf)(n)=0.\) Since \(n \in {\mathbb{Z}}\) is arbitrary, we see \(Xf=0.\) Since \(f \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2})\) is arbitrary, we see \(X=0.\) Namely, \(B=C,\) which shows that an operator \(A^{*}\) satisfying (186) is unique if it exists. Next, we construct an operator \(A^{*}\) satisfying (186). For \(f \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2})\) and \(n \in {\mathbb{Z}},\) we define \((A^{*}f)(n) \in {\mathbb{C}}^{2}\) by
where \(k_{A}(m,n)^{*}\) is the adjoint matrix of the matrix \(k_{A}(m,n).\) The sum in (190) is finite, since \(f \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2}).\) Thus, the formula (190) defines a vector \((A^{*}f)(n) \in {\mathbb{C}}^{2}.\) Therefore, we have a linear map \(A^{*} :C_{0}({\mathbb{Z}},{\mathbb{C}}^{2}) \rightarrow {\mathrm{Map}}\,({\mathbb{Z}},{\mathbb{C}}^{2}).\) For \(g \in C_{0}({\mathbb{Z}},{\mathbb{C}}^{2})\), the pairing \(\langle \,A^{*}f,g \,\rangle\) is defined and
which shows that \(A^{*}\) defined by (190) satisfies (186). \(\square\)
For example, the kernel function U(n, m) (\(n,m \in {\mathbb{Z}}\)) of the quantum walk \(U({\mathcal{C}})\) defined in (2) is given as
where \(\delta :{\mathbb{Z}} \rightarrow {\mathbb{C}}\) is defined by
The kernel function \(U^{*}(n,m)\) of the formal adjoint \(U({\mathcal{C}})^{*}\) is given as
From this, we have the formula (30).
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Tate, T. An eigenfunction expansion formula for one-dimensional two-state quantum walks. Ann. Funct. Anal. 13, 65 (2022). https://doi.org/10.1007/s43034-022-00210-8
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DOI: https://doi.org/10.1007/s43034-022-00210-8