A Rellich Type Theorem for Discrete Schr{\"o}dinger Operators

An analogue of Rellich's theorem is proved for discrete Laplacian on square lattice, and applied to show unique continuation property on certain domains as well as non-existence of embedded eigenvalues for discrete Schr{\"o}dinger operators.

Then u(x) = 0 on {|x| > R 0 }. This theorem has been extended to a broad class of Schrödinger operators, since it implies the non-existence of eigenvalues embedded in the continuous spectrum (see e.g. [14], [22], [1]), and also plays an important role in the proof of limiting absorption principle which yields the absolute continuity of the continuous spectrum (see e.g. [5], [11]). The Rellich type theorem states a local property at infinity of solutions. Namely, it proves u(x) = 0 on {|x| > R 1 } for some R 1 > R 0 . By the unique continuation property, it then follows that u(x) = 0 for |x| > R 0 . In the theory of linear partial differential equations (PDE), the Rellich type theorem can be regarded as the problem of division in the momentum space. In fact, given a linear PDE with constant coefficients P (D)u = f , f being compactly supported, Fourier transformation leads to the algebraic equation P (ξ) u(ξ) = f (ξ), where u(ξ) denotes the Fourier transform of u(x). If P (ξ) divides f (ξ), u is compactly supported due to the Paley-Wiener theorem. This approach was pursued by Treves [25], and then developed by Littman [17], [18], Hörmander [9] and Murata [19]. One should note that Besov spaces appear naturally through these works. In this paper, we shall consider its extension to the discrete case.
A precursor of this theorem is given in the proof of Theorem 9 of Shaban-Vainberg [23]. Their purpose is to compute the asymptotic expansion of the resolvent R 0 (λ ± i0) = (−∆ disc − λ ∓ i0) −1 on Z d and to find the associated radiation condition. Let be the standard d-dimensional torus and put Passing to the Fourier series, −∆ disc is transformed to the operator of multiplication by h(x) on T d . Therefore, the computation of the behavior of R 0 (λ±i0) boils down to that for an integral on M λ . For a compactly supported function f ∈ ℓ 2 (Z d ) and λ ∈ (0, d)\ E, E being a set of some exceptional points, the stationary phase method gives the following asymptotic expansion as |k| → ∞: where ω k = k/|k|, and the summation ranges over all stationary phase points x(λ, ω k ; j) ∈ M λ at which the normal of M λ is parallel to ω k and the Gaussian curvature does not vanish. It is then natural to define the radiation condition by using the first term of the above asymptotic expansion (1.5). To show the uniqueness of solutions to the discrete Schrödinger equation satisfying the radiation condition, they proved the assertion (which is buried in the proof actually) : (SV) The solution of (−∆ disc −λ) u = 0 on |n| > R 0 , satisfying u(n) = O(|n| −(d+1)/2 ) as |n| → ∞, vanishes on {|n| > R 1 } for large R 1 .
The new ingredient in the present paper is the following fact to be proved in §4: Consider the equation If the Fourier coefficients u(n) of the distribution u satisfy (1. 2) and f (n) is compactly supported, then u ∈ C ∞ (T d ), hence f (x) = 0 on M λ . Once we establish this fact, we can follow the arguments for proving the assertion (SV) without any change to show that u(n) is compactly supported. For the sake of completeness, in §4, we will also reproduce the proof of this part, which makes use of basic facts in theories of functions of several complex variables and algebraic geometry.
As applications of Theorem 1.1, we show in §2 non-existence of eigenvalues embedded in the continuous spectrum (except for threshold energies 0, 1, · · · , d) for −∆ disc + V in the whole space as well as in exterior domains.
The result of the present paper is used as a key step in [13] on the inverse scattering from the scattering matrix of a fixed energy for discrete Schrödinger operators with compactly supported potentials. In [12], the inverse scattering from all energies was studied by using complex Born approximation (see also [6]).
Function theory of several complex variables and algebraic geometry have already been utilized as powerful tools not only in linear PDE but also in the study of spectral properties for discrete Schrödinger operators or periodic problems. See e.g. Eskina [6], Kuchment-Vainberg [16], Gérard-Nier [7].
We give some remarks about notation in this paper. For x, y ∈ R d , x·y = x 1 y 1 + · · · + x d y d denotes the ordinary scalar product, and |x| = (x · x) 1/2 is the Euclidean norm. Note that even for n = (n 1 , · · · , n d ) ∈ Z d , we use |n| = ( d i=1 |n i | 2 ) 1/2 . For two Banach spaces X and Y , B(X, Y ) denotes the totality of bounded operators from X to Y . For a self-adjoint operator A on a Hilbert space, σ(A), σ ess (A), σ ac (A) and σ p (A) denote the spectrum, the essential spectrum, the absolutely continuous spectrum and the point spectrum of A, respectively. For a set S, # S denotes the number of elements in S. We use the notation where V is the multiplication operator : The assumption of the theorem yields σ ess ( H) = [0, d] (see [12]). Therefore, Theorem 2.1 asserts the non-existence of eigenvalues embedded in the continuous spectrum except for the set of thresholds Z ∩ [0, d].
where n ′ = (n 2 , · · · , n d ) and ∆ disc is the discrete Laplacian on Z d−1 . Repeating this procedure, we have u(n) = 0 for all n, and completes the proof.

2.2.
Unique continuation property. The next problem we address is the unique continuation property. We begin with the explanation of the exterior problem. A subset Ω ⊂ Z d is said to be connected if for any m, n ∈ Ω, there exists a sequence The interior • Ω and the boundary ∂Ω are defined by The normal derivative on ∂Ω is defined by Then, for a bounded connected subset Ω, the following Green formula holds: Indeed, the standard definition of Laplacian on graph is (see e.g. [4]) which yields Splitting the sum (2.6) into two parts, the ones over • Ω and over ∂Ω, we have (2.5). We take a connected exterior domain Ω ext , which means that there is a bounded set Ω int such that Ω ext = Z d \ Ω int , and consider the Schrödinger operator without imposing the boundary condition, where V is a real-valued compactly supported potential. Now suppose there exists a λ ∈ (0, d) \ Z, and u satisfying (1.2) and By Theorem 1.1, u vanishes near infinity. However, in the discrete case the unique continuation property of Laplacian does not hold in general. It depends on the shape of the domain. To guarantee it, we introduce the following cone condition. For 1 ≤ i ≤ d and n ∈ Z d , let C i,± (n) be the cone defined by Definition 2.
2. An exterior domain Ω ext is said to satisfy a cone condition if for any n ∈ Ω ext , there is a cone Examples of the domain satisfying this cone condition are • a domain with zigzag type boundary (see Figure 1).  Proof. Take any n ∈ Ω ext . By the cone condition, there is a cone, say C 1,+ (n), such that C 1,+ (n) ⊂ Ω ext . There is k 1 such that u(m) = 0 for m ∈ C 1,+ (n), k 1 < m 1 . Arguing as in the proof of Theorem 2.1, we have u(k 1 , m ′ ) = 0, (k 1 , m ′ ) ∈ C 1,+ (n). Repeating this procedure, we arrive at u(n) = 0.
An example of the domain which does not satisfy the cone condition is the one (in 2-dimension) whose boundary in the 4th quadrant has the form illustrated in Figure 2, and is rectangular in the other quadrants. In this case, u defined as in the figure satisfies ext ) = { u ∈ ℓ 2 (Ω ext ) ; ∂ ν u(n) + c(n) u(n) = 0, ∀n ∈ ∂Ω ext }, c(n) being a bounded real function on ∂D. They are bounded self-adjoint operators, and their essential spectra are Since this follows from the standard perturbation theory, we omit the proof. Theorem 2.3 asserts the non-exstence of embedded eigenvalues for these operators. In particular, it is clear from the proof that for the Dirichlet case, we have only to assume the cone condition for   Ωext, there is a cone C i,+ (n) or C i,− (n) such that C i,+ (n) ⊂ Ω ext or C i,− (n) ⊂ Ω ext .

Sobolev and Besov spaces on compact manifolds
The condition (1.2) is reformulated as a spectral condition for the Laplacian on the torus, which can further be rewritten by the Fourier transform. We do it on a compact Riemannian manifold in this section.
3.1. General case. Let M be a compact Riemannian manifold of dimension d. One way to introduce the Sobolev and Besov spaces on M is to use the Laplace- For s ∈ R, we define H s to be the completion of C ∞ (M ) by the norm L s/2 u , where · is the norm of L 2 (M ). We also define B * to be the completion of C ∞ (M ) by the norm Another way is to use the Fourier transform. Let {χ j } N j=1 be a partition of unity on M such that on each support of χ j , we can take one coordinate patch. We define H s to be the completion of C ∞ (M ) by the norm N j=1 ξ s (F χ j u)(ξ) , where F v = v denotes the Fourier transform of v, · is the norm of L 2 (R d ). We define B * to be the completion of C ∞ (M ) by the norm The following inclusion relations hold for s > 1/2: These definitions of Sobolev and Besov spaces coincide. We show Proof. We prove B * = B * . It is well-known that H s = H s for s ∈ R, whose proof is similar to, actually easier than, that for B * = B * given below.
First let us recall a formula from functional calculus. Let ψ(x) ∈ C ∞ (R) be such that for some m ∈ R. One can construct Ψ(z) ∈ C ∞ (C), called an almost analytic extension of ψ, having the following properties: In particular, if ψ(x) ∈ C ∞ 0 (R), one can take Ψ(z) ∈ C ∞ 0 (C). Then, if m < 0, for any self-adjoint operator A, we have the following formula which is called the formula of Helffer-Sjöstrand. See [8], [3], p. 390. We use a semi-classical analysis employing = 1/R as a small parameter (see e.g. [21]). We show that ψ( 2 L) is equal to, modulo a lower order term, a pseudo-differential operator (ΨDO) with symbol ψ(ℓ(x, ξ)), where ℓ(x, ξ) = d i,j=1 g ij (x)ξ i ξ j . In fact, take χ(x), χ 0 (x) ∈ C ∞ (M ) with small support such that χ 0 (x) = 1 on supp χ. Consider a ΨDO P (z) with symbol (ℓ(x, ξ) − z) −1 . Then S m being the standard Hörmander class of symbols (see [10], p. 65). This implies By the symbolic calculus, we have where N > 0 is a constant depending on s and d, and C s,d does not depend on . We take ψ ∈ C ∞ 0 (R) and apply (3.4). Then we have where Ψ is a ΨDO with symbol ψ(ℓ(x, ξ)) and We then have, letting z = x + iy, we obtain, using (3.6), where we have used (3.3) with m < −1, n = N + 1. This estimate implies In fact, taking 0 ≤ t ≤ 3/2, we have choosing s = −3 in (3.9), The right-hand side is bounded if 1/4 < t ≤ 3/4. Now, by the definition of B * , we have the following equivalence In fact, the left-hand side is equivalent to the right-hand side for one fixed ψ such that ψ(t) = 1 for |t| < 1, and ψ(t) = 0 for |t| > 2. By virtue of (3.7) and (3.10), (3.11) is equivalent to The symbol of (Ψ ) * χ 0 (x) 2 Ψ is equal to Then by a suitable choice of 0 < c 1 < c 2 , we have Moreover, we can assume that there exists q(x, ξ) ∈ C ∞ (R d × R d ) such that is equal to Then we have (3.14) In the right-hand side, Q 0, is a ΨDO with symbol q(x, ξ), where q(x, ξ) is given by (3.13), and Q 1, is a ΨDO with symbol q 1 (x, ξ; ) admitting the asymptotic expansion with q j (x, ξ) having the same support property as in (3.13). Since the 1st term of the right-hand side of (3.14) is non-negative, we have proven By a similar computation, we can prove By (3.16) and (3.17), we have where Q 1, and Q ′ 1, have the property (3.15). As u ∈ H −s , ∀s > 1/2, we have sup Therefore, by (3.18), sup for some c > 0, which is equivalent to u ∈ B * . We have thus completed the proof of Lemma 3.1.
By the same argument, we can also prove the following lemma.
Lemma 3.2. If u ∈ B * , we have the following equivalence for any j and any ψ ∈ C ∞ 0 (R), where {χ j } N j=1 is the partition of unity on M .

3.2.
Torus. We interprete the above results for the case of the torus T d defined by (1.3). Let U be the unitary operator from ℓ 2 (Z d ) to L 2 (T d ) defined by Letting We define operators N j and N j by We put N = (N 1 , · · · , N d ), and let N 2 be the self-adjont operator defined by where ∆ denotes the Laplacian on T d = [−π, π] d with periodic boundary condition. We can then apply the results in the previous subsection to L = −∆. We put For s ∈ R, let H s be the completion of D(|N | s ) with respect to the norm u s = N s u i.e.
For a self-adjoint operator T , let χ(a ≤ T < b) denote the operator χ I (T ), where χ I (λ) is the characteristic function of the interval I = [a, b). The operators χ(T < a) and χ(T ≥ b) are defined similarly. Using the series {r j } ∞ j=0 with r −1 = 0, r j = 2 j (j ≥ 0), we define the Besov space B by Its dual space B * is the completion of H by the following norm The following Lemma 3.3 is proved in the same way as in [2]. Lemma 3.3. (1) There exists a constant C > 0 such that (2) For s > 1/2, the following inclusion relations hold: In view of the above lemma, in the following, we use as a norm on B * . We also put H = ℓ 2 (Z d ), and define H s , B, B * by replacing N by N . Note that H s = U * H s and so on. In particular, Parseval's formula implies that u(n) being the Fourier coefficient of u(x).

Proof of Theorem 1.1
We extend u(n) to be zero for |n| ≤ R 0 and denote it by u again. Then we have where f is compactly supported. In fact, letting P (k) be the projection onto the site k, it is written as f = |k|≤R0+1 c k P (k) u. We first note the following Lemma.
By (4.1), u satisfies where f is a polynomial of e ixj , j = 1, · · · , d, since f is compactly supported. Take a point x (0) ∈ M λ , and let χ ∈ C ∞ (T d ) be such that χ(x (0) ) = 1 and supp χ is sufficiently small. Letting v(x) = χ(x)u(x), g(x) = χ(x)f (x) and making the change of variable x → y so that y 1 = h(x) − λ, we have by passing to the Fourier transform, ∂ ∂η1 v(η) = −i g(η). Integrating this equation, we have Since g(η) is rapidly decreasing, we then see the existence of the limit We show that this limit vanishes. Let D R be the slab such that Then we have D R ⊂ {|η| < R} for a sufficiently small δ > 0. We then see that As R → ∞, the right-hand side tends to zero by (4.4), hence so does the left-hand side, which proves that lim η1→∞ v(η) = 0. We have, therefore, This shows that v = χu ∈ C ∞ (T d ).
It is easy to see that u is smooth outside M λ . Then u ∈ C ∞ (T d ).
We use the function theory of several complex variables. Let T d C = C d /(2πZ) d be the complex torus and define Lemma 4.2. For λ ∈ (0, d) \ Z, M C λ is a (d − 1)-dimensional, connected complex submanifold of T d C .
Hence F (w)/H λ (w) is analytic except only on some sets of complex dimension d− 2 (the intersection of two hyperplanes). Therefore, • F (w)/H λ (w) is an entire function.
Finally, we use the following fact, a corollary of the Hilbert Nullstellensatz (See e.g. Appendix 6 of [24]). Let C[w 1 , · · · , w d ] be the ring of polynomials of variables w 1 , · · · , w d .
Lemma 4.4. If f, g ∈ C[w 1 , · · · , w d ], and suppose that f is irreducible. If g = 0 on all zeros of f , there exists h ∈ C[w 1 , · · · , w d ] such that g = f h.