Generalized Indefinite Strings with Purely Discrete Spectrum

Abstract

We establish criteria for the spectrum of a generalized indefinite string to be purely discrete and to satisfy Schatten–von Neumann properties. The results can be applied to the isospectral problem associated with the conservative Camassa–Holm flow and to Schrödinger operators with $\delta '$-interactions.

Dedicated to the memory of Sergey Nikolaevich Naboko (1950–2020)

Appendices

Appendix A: On a Class of Integral Operators

Let $\mathsf {q}$ be a function in $L^2_{{\mathrm {loc}}}[0,\infty )$ and consider the integral operator $\mathrm {J}$ in the Hilbert space $L^2[0,\infty )$ defined by

$$\displaystyle \begin{aligned} {} \mathrm{J} f(x) = \int_0^\infty \mathsf{q}(\max(x,t))f(t)dt = \mathsf{q}(x)\int_0^x f(t)dt + \int_x^\infty \mathsf{q}(t)f(t)dt \end{aligned} $$

(A.1)

for functions $f\in \dot {L}^2_{{\mathrm {c}}}[0,\infty )$. Since the subspace $\dot {L}^2_{{\mathrm {c}}}[0,\infty )$ is dense in $L^2[0,\infty )$, the operator $\mathrm {J}$ is densely defined. The theorems in this appendix gather a number of results from [2] for these kinds of integral operators.

Theorem A.1

The following assertions hold true:

• (i) The operator$\mathrm {J}$is bounded if and only if there is a constant$c\in {\mathbb C}$such that

  $$\displaystyle \begin{aligned} {} \limsup_{x\rightarrow\infty}\, x\int_x^\infty |\mathsf{q}(t) - c|{}^2dt <\infty. \end{aligned} $$

  (A.2)

  In this case, the constant c is given by

  $$\displaystyle \begin{aligned} {} c = \lim_{x\rightarrow \infty} \frac{1}{x} \int_0^{x} \mathsf{q}(t)dt. \end{aligned} $$

  (A.3)

• (ii) The operator$\mathrm {J}$is compact if and only if there is a constant$c\in {\mathbb C}$such that

  $$\displaystyle \begin{aligned} {} \lim_{x\rightarrow \infty} x\int_x^\infty |\mathsf{q}(t) - c|{}^2dt =0. \end{aligned} $$

  (A.4)

• (iii) For each$p>1$, the operator$\mathrm {J}$belongs to the Schatten–von Neumann class$\mathfrak {S}_p$if and only if there is a constant$c\in {\mathbb C}$such that

  $$\displaystyle \begin{aligned} {} \int_0^\infty \biggl(x\int_x^\infty |\mathsf{q}(t) - c|{}^2dt\biggr)^{{p}/{2}} \frac{dx}{x} < \infty. \end{aligned} $$

  (A.5)

• (iv) If the operator$\mathrm {J}$belongs to the Hilbert–Schmidt class$\mathfrak {S}_2$, then its Hilbert–Schmidt norm is given by

  $$\displaystyle \begin{aligned} {} \|\mathrm{J}\|{}^2_{\mathfrak{S}_2} = 2\int_0^\infty x |\mathsf{q}(x) - c|{}^2 dx, \end{aligned} $$

  (A.6)

  where the constant c is given by (A.3).

• (v) If the operator$\mathrm {J}$belongs to the trace class$\mathfrak {S}_1$, then

  $$\displaystyle \begin{aligned} {} \int_0^\infty \biggl(x\int_x^\infty |\mathsf{q}(t) - c|{}^2dt\biggr)^{{1}/{2}} \frac{dx}{x} <\infty, \end{aligned} $$

  (A.7)

  the function$\mathsf {q}-c$is integrable and the trace of$\mathrm {J}$is given by

  $$\displaystyle \begin{aligned} {} \mathrm{tr}\,\mathrm{J} = \int_0^\infty (\mathsf{q}(x)-c) dx, \end{aligned} $$

  (A.8)

  where the constant c is given by (A.3).

Proof

Sufficiency of the conditions in (i), (ii), and (iii) follows readily from [2, Section 3] upon noticing that one has

$$\displaystyle \begin{aligned} \mathrm{J} f(x) = \int_0^\infty \mathsf{q}(\max(x,t)) f(t)dt = \int_0^\infty (\mathsf{q}(\max(x,t)) - c)f(t)dt \end{aligned}$$

for functions $f \in \dot {L}^2_{{\mathrm {c}}}[0,\infty )$. In order to prove that the condition in (i) is also necessary, let us suppose that the operator $\mathrm {J}$ is bounded. For every $n\in {\mathbb N}$, we consider the function

$$\displaystyle \begin{aligned} f_n = {\mathbb{1}}_{[0,1)} - \frac{1}{n}{\mathbb{1}}_{[1,n+1)}, \end{aligned}$$

where ${\mathbb{1} }_I$ denotes the characteristic function of an interval $I\subseteq [0,\infty )$. Clearly, the functions $f_n$ belong to $\dot {L}^2_{{\mathrm {c}}}[0,\infty )$ and converge to ${\mathbb{1} }_{[0,1)}$ in $L^2[0,\infty )$ as $n\rightarrow \infty$. Since the operator $\mathrm {J}$ is bounded, this implies that the functions $\mathrm {J} f_n$ converge in $L^2[0,\infty )$. In view of the definition of $\mathrm {J}$, we then find that

$$\displaystyle \begin{aligned} \mathrm{J} f_n(x) &=\begin{cases} x\mathsf{q}(x) + \int_x^1 \mathsf{q}(t)dt - \frac{1}{n}\int_1^{n+1}\mathsf{q}(t)dt, & x\in [0,1),\\[2mm] \mathsf{q}(x) - \frac{1}{n}\mathsf{q}(x)(x-1) - \frac{1}{n}\int_x^{n+1}\mathsf{q}(t)dt, & x\in [1,n+1),\\[2mm] 0, & x\in[n+1,\infty), \end{cases} \end{aligned} $$

for almost all $x\in [0,\infty )$. From this we are able to infer that the limit

$$\displaystyle \begin{aligned} c := \lim_{n\rightarrow \infty} \frac{1}{n} \int_1^{n+1} \mathsf{q}(t)dt = \lim_{n\rightarrow \infty}\langle Q - \mathrm{J} f_n , {\mathbb{1}}_{[0,1)} \rangle _{L^2[0,\infty)} \end{aligned} $$

exists in ${\mathbb C}$, where the function Q in $L^2[0,\infty )$ is defined by

$$\displaystyle \begin{aligned} Q(x) = {\mathbb{1}}_{[0,1)}(x)\biggl( x\mathsf{q}(x) + \int_x^1 \mathsf{q}(t)dt\biggr). \end{aligned} $$

Moreover, we see that for almost every $x\in [0,\infty )$ one has

$$\displaystyle \begin{aligned} \lim_{n\rightarrow \infty} \mathrm{J} f_n(x) = \begin{cases} Q(x) - c, & x\in [0,1),\\ \mathsf{q}(x) - c, & x\in[1,\infty). \end{cases} \end{aligned}$$

Since, on the other side, it can be readily checked that one also has

$$\displaystyle \begin{aligned} \int_0^\infty (\mathsf{q}(\max(x,t)) - c){\mathbb{1}}_{[0,1)}(t)dt = \begin{cases} Q(x) - c, & x\in [0,1),\\ \mathsf{q}(x) - c, & x\in[1,\infty), \end{cases} \end{aligned}$$

we may conclude that the bounded extension of $\mathrm {J}$ to $L^2_{\mathrm {c}}[0,\infty )$ satisfies

$$\displaystyle \begin{aligned} \mathrm{J} {\mathbb{1}}_{[0,1)}(x) = \int_0^\infty (\mathsf{q}(\max(x,t)) - c){\mathbb{1}}_{[0,1)}(t)dt \end{aligned} $$

and is hence explicitly given by

$$\displaystyle \begin{aligned} \mathrm{J} f(x) = \int_0^\infty (\mathsf{q}(\max(x,t)) - c)f(t)dt \end{aligned} $$

for all functions $f \in L^2_{{\mathrm {c}}}[0,\infty )$. It now remains to apply [2, Theorem 3.1] to deduce that the function $\mathsf {q}$ satisfies (A.2) and the constant c is necessarily given by (A.3). If the operator $\mathrm {J}$ is moreover compact, then [2, Theorem 3.2] yields (A.4) and if it belongs to $\mathfrak {S}_p$ for some $p>1$, then [2, Theorem 3.3] yields (A.5). This proves that the conditions in (ii) and (iii) are also necessary. Finally, the formula (A.6) for the Hilbert–Schmidt norm in (iv) follows from [2, Remark in Section 3] and the necessary condition (A.7) for the operator $\mathrm {J}$ to belong to the trace class $\mathfrak {S}_1$ as well as the formula (A.8) for the trace in (v) follow from [2, Theorem 6.2]. □

Remark A.2

Let us stress that boundedness and compactness criteria for the integral operator $\mathrm {J}$ have been established before in [6] and [33], respectively.

In Theorem A.1(iii), the value $p=1$ is a threshold since the condition (A.7) is only necessary for the operator $\mathrm {J}$ to belong to the trace class $\mathfrak {S}_1$; see [2, Section 6].

Theorem A.3

Let$\chi$be a non-negative Borel measure on$[0,\infty )$and suppose that

$$\displaystyle \begin{aligned} \mathsf{q}(x) = \int_{[0,x)}d\chi \end{aligned} $$

(A.9)

for almost all$x\in [0,\infty )$. The following assertions hold true:

• (i) The operator$\mathrm {J}$is bounded if and only if

  $$\displaystyle \begin{aligned} {} \limsup_{x\rightarrow\infty}\, x \int_{[x,\infty)}d\chi <\infty. \end{aligned} $$

  (A.10)

• (ii) The operator$\mathrm {J}$is compact if and only if

  $$\displaystyle \begin{aligned} {} \lim_{x\rightarrow \infty} x \int_{[x,\infty)}d\chi =0. \end{aligned} $$

  (A.11)

• (iii) For each$p>{1}/{2}$, the operator$\mathrm {J}$belongs to the Schatten–von Neumann class$\mathfrak {S}_p$if and only if

  $$\displaystyle \begin{aligned} {} \int_0^\infty \biggl(x\int_{[x,\infty)}d\chi\biggr)^p\frac{dx}{x} <\infty. \end{aligned} $$

  (A.12)

• (iv) If the operator$\mathrm {J}$belongs to the trace class$\mathfrak {S}_1$, then its trace is given by

  $$\displaystyle \begin{aligned} \mathrm{tr}\,\mathrm{J} = -\int_{[0,\infty)} x\, d\chi(x). \end{aligned} $$

  (A.13)

• (v) If the operator$\mathrm {J}$belongs to the Schatten–von Neumann class$\mathfrak {S}_{{1}/{2}}$, then the measure$\chi$is singular with respect to the Lebesgue measure.

Proof

By means of the connection established in the proof of Theorem A.1, the claims in (i), (ii) and (iii) follow from [2, Theorem 4.6], upon also noting that

$$\displaystyle \begin{aligned} \lim_{x\rightarrow \infty}\frac{1}{x}\int_0^{x} \mathsf{q}(t)dt = \lim_{x\rightarrow \infty} \int_{[0,x)} \biggl(1-\frac{t}{x}\biggr) d\chi(t) = \int_{[0,\infty)} d\chi \end{aligned}$$

in this case. The claim in (iv) then follows from Theorem A.1(v) and the claim in (v) follows from [2, Corollary 8.12]. □

We also want to consider related operators on a finite interval. To this end, let L be a positive number and define the operator $\mathrm {J}_L$ in the Hilbert space $L^2[0,L)$ by

$$\displaystyle \begin{aligned} {} \mathrm{J}_L f(x) = \int_0^L \mathsf{q}_L(\min(x,t))f(t)dt = \int_0^x \mathsf{q}_L(t)f(t)dt + \mathsf{q}_L(x) \int_x^L f(t)dt \end{aligned} $$

(A.14)

for functions $f\in L^2_{{\mathrm {c}}}[0,L)$, where $\mathsf {q}_L$ is a function in $L^2_{{\mathrm {loc}}}[0,L)$. As a Carleman integral operator, the operator $\mathrm {J}_L$ is closable (see [16, Theorem 3.8] for example).

Theorem A.4

The following assertions hold true:

• (i) The operator$\mathrm {J}_L$is bounded if and only if

  $$\displaystyle \begin{aligned} \limsup_{x\rightarrow L}\, (L-x)\int_0^x |\mathsf{q}_L(t)|{}^2dt < \infty. \end{aligned} $$

  (A.15)

• (ii) The operator$\mathrm {J}_L$is compact if and only if

  $$\displaystyle \begin{aligned} \lim_{x\rightarrow L} (L-x)\int_0^x |\mathsf{q}_L(t)|{}^2dt = 0. \end{aligned} $$

  (A.16)

• (iii) For each$p>1$, the operator$\mathrm {J}_L$belongs to the Schatten–von Neumann class$\mathfrak {S}_p$if and only if

  $$\displaystyle \begin{aligned} \int_0^L \biggl((L-x)\int_0^x |\mathsf{q}_L(t)|{}^2dt\biggr)^{{p}/{2}}\frac{dx}{L-x} < \infty. \end{aligned} $$

  (A.17)

• (iv) If the operator$\mathrm {J}_L$belongs to the Hilbert–Schmidt class$\mathfrak {S}_2$, then its Hilbert–Schmidt norm is given by

  $$\displaystyle \begin{aligned} \|\mathrm{J}_L\|{}^2_{\mathfrak{S}_2} = 2 \int_0^L (L-x) |\mathsf{q}_L(x)|{}^2 dx. \end{aligned} $$

  (A.18)

• (v) If the operator$\mathrm {J}_L$belongs to the trace class$\mathfrak {S}_1$, then

  $$\displaystyle \begin{aligned} {} \int_0^L \biggl((L-x)\int_0^x |\mathsf{q}_L(t)|{}^2dt\biggr)^{{1}/{2}}\frac{dx}{L-x} < \infty, \end{aligned} $$

  (A.19)

  the function$\mathsf {q}_L$is integrable and the trace of$\mathrm {J}_L$is given by

  $$\displaystyle \begin{aligned} {} \mathrm{tr}\,\mathrm{J}_L = \int_0^L \mathsf{q}_L(x)dx. \end{aligned} $$

  (A.20)

Proof

We first observe that for functions $f\in L^2_{{\mathrm {c}}}[0,L)$ one has

$$\displaystyle \begin{aligned} \mathrm{J}_L f(L-x) & = \int_0^L \mathsf{q}_L(\min(L-x,t))f(t)dt \\ &= \int_0^L \mathsf{q}_L(\min(L-x,L-t))f(L-t)dt \\ &= \int_0^L \mathsf{q}_L(L - \max(x,t))f(L-t)dt \end{aligned} $$

for almost all $x\in (0,L)$. Taking into account that the map $f \mapsto f(L-\,\cdot )$ is unitary on $L^2[0,L)$, the claims in (i), (ii), and (iii) follow readily from the corresponding results in [2, Section 3] with the function $\varphi$ in $L^2_{{\mathrm {loc}}}(0,\infty )$ given by

$$\displaystyle \begin{aligned} \varphi(x)= \begin{cases} \mathsf{q}_L(L-x), & x\in (0,L), \\ 0, & x\in[L,\infty). \end{cases} \end{aligned}$$

The claims in (iv) and (v) then follow from [2, Remark in Section 3] and [2, Theorem 6.2], respectively. □

The value $p=1$ is again a threshold in Theorem A.4(iii) because the condition (A.19) is only necessary for the operator $\mathrm {J}_L$ to belong to the trace class.

Theorem A.5

Let$\chi$be a non-negative Borel measure on$[0,L)$and suppose that

$$\displaystyle \begin{aligned} \mathsf{q}_L(x) = \int_{[0,x)}d\chi \end{aligned} $$

(A.21)

for almost all$x\in [0,L)$. The following assertions hold true:

• (i) The operator$\mathrm {J}_L$is bounded if and only if

  $$\displaystyle \begin{aligned} \limsup_{x\rightarrow L}\, (L-x)\int_{[0,x)}d\chi <\infty. \end{aligned} $$

  (A.22)

• (ii) The operator$\mathrm {J}_L$is compact if and only if

  $$\displaystyle \begin{aligned} \lim_{x\rightarrow L} (L-x)\int_{[0,x)}d\chi =0. \end{aligned} $$

  (A.23)

• (iii) For each$p>{1}/{2}$, the operator$\mathrm {J}_L$belongs to the Schatten–von Neumann class$\mathfrak {S}_p$if and only if

  $$\displaystyle \begin{aligned} \int_0^L \biggl((L-x)\int_{[0,x)}d\chi\biggr)^p\frac{dx}{L-x} <\infty. \end{aligned} $$

  (A.24)

• (iv) If the operator$\mathrm {J}_L$belongs to the trace class$\mathfrak {S}_1$, then its trace is given by

  $$\displaystyle \begin{aligned} \mathrm{tr}\,\mathrm{J}_L = \int_{[0,L)} (L-x) d\chi(x). \end{aligned} $$

  (A.25)

• (v) If the operator$\mathrm {J}_L$belongs to the Schatten–von Neumann class$\mathfrak {S}_{{1}/{2}}$, then the measure$\chi$is singular with respect to the Lebesgue measure.

Proof

By means of the connection established in the proof of Theorem A.4, the claims in (i), (ii) and (iii) follow from [2, Theorem 4.6], the claim in (iv) follows from Theorem A.4(v) and the claim in (v) follows from [2, Corollary 8.12]. □

Appendix B: Linear Relations

Let $\mathcal {H}$ be a separable Hilbert space. A (closed) linear relation in $\mathcal {H}$ is a (closed) linear subspace of $\mathcal {H}\times \mathcal {H}$. Since every linear operator in $\mathcal {H}$ can be identified with its graph, the set of linear operators can be regarded as a subset of all linear relations in $\mathcal {H}$. Recall that the domain, the range, the kernel and the multi-valued part of a linear relation $\Theta$ are given, respectively, by

$$\displaystyle \begin{aligned} \mathrm{dom}(\Theta) &= \{f\in \mathcal{H}\,|\, \exists g\in\mathcal{H}\ \text{such that}\ (f,g)\in \Theta\}, \end{aligned} $$

(B.1)

$$\displaystyle \begin{aligned} \mathrm{ran}(\Theta) &= \{g\in \mathcal{H}\,|\, \exists f\in\mathcal{H}\ \text{such that}\ (f,g)\in \Theta\}, \end{aligned} $$

(B.2)

$$\displaystyle \begin{aligned} \mathrm{ker}(\Theta) &= \{f\in \mathcal{H}\,|\, (f,0)\in \Theta\}, \end{aligned} $$

(B.3)

$$\displaystyle \begin{aligned} \mathrm{mul}(\Theta) &= \{g\in \mathcal{H}\,|\, (0,g)\in \Theta\}. \end{aligned} $$

(B.4)

The adjoint linear relation $\Theta ^\ast$ of a linear relation $\Theta$ is defined by

$$\displaystyle \begin{aligned} \Theta^\ast = \big\{ (\tilde{f},\tilde{g})\in \mathcal{H}\times\mathcal{H}\,|\, \langle g , \tilde{f} \rangle _{\mathcal{H}} = \langle f , \tilde{g} \rangle _{\mathcal{H}}\ \text{for all}\ (f,g)\in\Theta\big\}. \end{aligned} $$

(B.5)

The linear relation $\Theta$ is called symmetric if $\Theta \subseteq \Theta ^\ast$. It is called self-adjoint if $\Theta =\Theta ^\ast$. Note that $\mathrm {mul}(\Theta )$ is orthogonal to $\mathrm {dom}(\Theta )$ if $\Theta$ is symmetric. For a closed symmetric linear relation $\Theta$ satisfying $\mathrm {mul}(\Theta ) = \mathrm {mul}(\Theta ^\ast )$ (the latter is further equivalent to the fact that $\Theta$ is densely defined on $\mathrm {mul}(\Theta )^\perp$), setting

$$\displaystyle \begin{aligned} \mathcal{H}_{\mathrm{op}}=\overline{\mathrm{dom}(\Theta)} = \mathrm{mul}(\Theta)^\perp, \end{aligned} $$

(B.6)

we obtain the following orthogonal decomposition

$$\displaystyle \begin{aligned} {} \Theta = \Theta_{\mathrm{op}}\oplus \Theta_{\infty}, \end{aligned} $$

(B.7)

where $\Theta _\infty = \{0\}\times \mathrm {mul}(\Theta )$ and $\Theta _{\mathrm {op}}$ is the graph of a closed symmetric linear operator in $\mathcal {H}_{\mathrm {op}}$, called the operator part of $\Theta$. Notice that for non-closed symmetric linear relations, the decomposition (B.7) may not hold true.

If $\Theta _1$ and $\Theta _2$ are linear relations in $\mathcal {H}$, then their sum $\Theta _1+\Theta _2$ and their product $\Theta _2\Theta _1$ are defined by

$$\displaystyle \begin{aligned} \Theta_1+\Theta_2 & = \{(f,g_1+g_2)\,|\, (f,g_1)\in\Theta_1, \ (f,g_2)\in\Theta_2\}, \end{aligned} $$

(B.8)

$$\displaystyle \begin{aligned} \Theta_2\Theta_1 & = \{(f,g)\,|\, (f,h)\in\Theta_1, \ (h,g)\in\Theta_2\ \text{for some}\ h\in\mathcal{H}\}. \end{aligned} $$

(B.9)

The inverse of a linear relation $\Theta$ is given by

$$\displaystyle \begin{aligned} \Theta^{-1} = \{(g,f)\in \mathcal{H}\times\mathcal{H} \,|\, (f,g)\in \Theta\}. \end{aligned} $$

(B.10)

Consequently, one can consider $(\Theta - z)^{-1}$ for any $z\in {\mathbb C}$. The set of those $z\in {\mathbb C}$ for which $(\Theta - z)^{-1}$ is the graph of a closed bounded operator on $\mathcal {H}$ is called the resolvent set of $\Theta$ and denoted by $\rho (\Theta )$. Its complement $\sigma (\Theta )={\mathbb C}\backslash \rho (\Theta )$ is called the spectrum of $\Theta$. If $\Theta$ is self-adjoint, then taking into account (B.7) we obtain

$$\displaystyle \begin{aligned} {} (\Theta - z)^{-1} = (\Theta_{\mathrm{op}} - z)^{-1}\oplus \mathbb{O}_{\mathrm{mul}(\Theta)}. \end{aligned} $$

(B.11)

This immediately implies that $\rho (\Theta ) = \rho (\Theta _{\mathrm {op}})$, $\sigma (\Theta ) = \sigma (\Theta _{\mathrm {op}})$ and, moreover, one can introduce the spectral types of $\Theta$ as those of its operator part $\Theta _{\mathrm {op}}$.