1 Introduction and main results

Let \({\mathfrak {H}}=\ell ^2({\mathbb {Z}}_+) \) be the Hilbert space of square summable sequences on \({\mathbb {Z}}_+=\{1,2,3,\ldots \}\). Let \(V:\,{\mathfrak {H}}\mapsto {\mathfrak {H}}\) be the operator of multiplication by a bounded complex-valued function on \({\mathbb {Z}}_+\). We study the spectral properties of the Schrödinger operator H, defined in \({\mathfrak {H}}\) by

$$\begin{aligned} (H\,u )_j=\sum _{|l-j|=1}u_l+V_ju_j,\quad \forall j\ge 2. \end{aligned}$$
(1.1)

Additionally, we set

$$\begin{aligned} (Hu)_1=u_2+V_1u_1. \end{aligned}$$

Note that H is a bounded operator. The spectrum of the self-adjoint operator \(H_0=H-V\) coincides with the interval \([-2,2]\) and is absolutely continuous. Let \(\lambda _j\) denote the eigenvalues of the operator (1.1). We are interested in an estimate of the total number N of eigenvalues \(\lambda _j\) in the case where the sequence \(V_j\) decays exponentially fast.

More precisely, we shall prove the following two theorems:

Theorem 1.1

The number N of eigenvalues of H in \(\ell ^2({\mathbb {Z}}_+)\), counting algebraic multiplicities, satisfies

$$\begin{aligned} N \le \frac{1}{2\ln \Lambda } \left( \frac{2\Lambda ^2}{\Lambda ^2-1}\sum _{n=1}^\infty \Lambda ^{2 n} |V_n| \right) ^2 , \end{aligned}$$

for any \(\Lambda >1\).

A similar result for a continuous operator was proved in [13] by Frank, Laptev and Safronov.

We also establish a slightly different estimate:

Theorem 1.2

The number N of eigenvalues of H in \(\ell ^2({\mathbb {Z}}_+)\), counting algebraic multiplicities, satisfies

$$\begin{aligned} N \le \frac{1}{\ln \Lambda } \frac{\Lambda ^2}{(\Lambda ^2-1)} \left( \sum _{n=1}^\infty \Lambda ^{ n} |V_n|^{1/2} \right) ^2 , \end{aligned}$$

for any \(\Lambda >1\).

Note that the right hand sides of both estimates can be finite only in the case where V is an exponentially decaying potential. It turns out that N might be finite even in the case when the potential decays slower. For instance, the operator \(-d^2/dx^2+V(x)\) on the half-line \([0,\infty )\) has finitely many eigenvalues if \(|V|\le C\exp (-c\sqrt{x})\) for some \(C,\, c>0\). This remarkable result was proved by Pavlov in [23]. It was established that the eigenvalues can not accumulate to a point of the positive half- line, which is enough to conclute that the set of all eigenvalues is finite.

On the other hand, there is another remarkable result of Pavlov (see [24]), which says that, for any \(0<p<1/2\), there exists a complex-valued potential V satisfying \(|V|\le C\exp (-c| x|^p)\) and a complex number \(\theta \), such that the operator \(-d^2/dx^2+V(x)\) with the boundary condition \(\psi '(0)=\theta \psi (0)\) has infinitely many eigenvalues. Another interesting result was recently established by Bögli [2]. It was shown that there exists a potential for which the eigenvalues accumulate to every point on \([0,\infty )\).

2 Zeroes of analytic functions

The following proposition gives a useful bound on the zeroes of an analytic function in the compliment of the disc of radius \(R>0\).

Proposition 2.1

Let \(0<R<1\). Let \(a(\cdot )\) be an analytic function in \(\{k:\, |k|>R\}\). Assume that \(a(\cdot )\) is continuous up to the boundary and satisfies

$$\begin{aligned} a(k) = 1 + O(|k|^{-1}) \quad \text {as}\ |k|\rightarrow \infty \ \text {in}\ \{k:\,|k|>R\}. \end{aligned}$$
(2.1)

Assume also that for some \(A\ge 1\),

$$\begin{aligned} |a(k)| \le A , \quad \text {if}\ |k |= R\,. \end{aligned}$$
(2.2)

Then the zeroes \(k_j\) of \(a(\cdot )\) in \(\{k:\,|k|>R\}\), repeated according to their multiplicities, satisfy

$$\begin{aligned} \prod _j \left( \frac{ |k_j |}{ R }\right) \le A . \end{aligned}$$
(2.3)

Proof

We introduce the Blaschke product

$$\begin{aligned} B(k) = \prod _j \frac{k-k_j}{R-R^{-1}\overline{k_j} k}\,. \end{aligned}$$

Clearly, a(k) / B(k) is an analytic and non-zero in \(\{k:\, |k|>R\}\). Consequently, \(\log (a(k)/B(k))\) exists and is analytic in \( \{k:\,|k|>R\}\). Let \(C_R\) denote the circle \(\{k\in {\mathbb {C}}:\, |k|=R \}\), traversed counterclockwise.

Then, according to the residue calculus,

$$\begin{aligned} \int _{C_R} \log \frac{a(k)}{B(k)}\,\frac{dk}{k} = 2\pi i \,\,\lim _{k\rightarrow \infty } \log \frac{a(k)}{B(k)}=2\pi i\,\,\sum _j \log \frac{\bar{k}_j}{-R}\,, \end{aligned}$$

and therefore

$$\begin{aligned} \int _{-\pi }^\pi \log \frac{a(Re^{i\varphi })}{B(Re^{i\varphi })}\,d\varphi =2\pi \,\,\sum _j \log \frac{\bar{k}_j}{-R}\,. \end{aligned}$$
(2.4)

We note that \(|B(Re^{i\varphi })|=1\) if \(\varphi \in \mathbb {R}\) and, therefore,

$$\begin{aligned} {{\mathrm{Re}}}\int _{-\pi }^\pi \log \frac{a(Re^{i\varphi })}{B(Re^{i\varphi })}\,d\varphi = \int _{-\pi }^\pi \ln \Bigl | \frac{a(Re^{i\varphi })}{B(Re^{i\varphi })}\Bigr |\,d\varphi = \int _{-\pi }^\pi \ln |a(Re^{i\varphi })|\,d\varphi . \end{aligned}$$
(2.5)

On the other hand,

$$\begin{aligned} {{\mathrm{Re}}}\,\,\sum _j \log \frac{\bar{k}_j}{-R} = \,\,\sum _j \ln \frac{|k_j|}{R}. \end{aligned}$$
(2.6)

We conclude from (2.4), (2.5) and (2.6) that

$$\begin{aligned} \int _{-\pi }^\pi \ln |a(Re^{i\varphi })|\,d\varphi = 2\pi \,\,\sum _j \ln \frac{|k_j|}{R}. \end{aligned}$$
(2.7)

Finally, by (2.2),

$$\begin{aligned} \int _{-\pi }^\pi \ln |a(Re^{i\varphi })|\,d\varphi&\le 2\pi \ln A,. \end{aligned}$$
(2.8)

Inequality (2.3) now follows from (2.7) and (2.8). \(\square \)

Corollary 2.2

Let \(0<R<1\). Let \(a(\cdot )\) be an analytic function in \(\{k:\, |k|>R\}\) satisfying (2.1). Assume that, for any \(R'>R\) sufficiently close to R, condition (2.2) holds with R replaced by \(R'\). Then the number

$$\begin{aligned} {\mathcal N}:=\#\{ j:\,|k_j|\ge 1 \} \end{aligned}$$

of zeroes \(k_j\) of \(a(\cdot )\) in \(\{ k:\, |k|\ge 1\}\), repeated according to their multiplicities, satisfies

$$\begin{aligned} {\mathcal N} \le \frac{\ln A}{\ln 1/R} \,. \end{aligned}$$

Proof

We apply Proposition 2.1 for every \(R'>R\) sufficiently close to R and obtain

$$\begin{aligned} \sum _j \left( \ln |k_j |-\ln R' \right) _+ \le \ln A \end{aligned}$$

Clearly, we have

$$\begin{aligned} \sum _j \left( \ln |k_j |-\ln R'\right) _+ \ge |\ln R'|\cdot \#\left\{ j:\, |k_j|\ge 1 \right\} \,. \end{aligned}$$

Consequently,

$$\begin{aligned} |\ln R'| \cdot {\mathcal N}\le \ln A. \end{aligned}$$

The corollary follows by passing to the limit \(R'\rightarrow R\). \(\square \)

3 Classes of compact operators and determinants

Let \(1\le p<\infty \). We say that a compact operator T belongs to the Schatten class \({\mathfrak {S}}_p\) if its singular values \(s_j(T)\) satisfy

$$\begin{aligned} \Vert T\Vert ^p_{{\mathfrak {S}}_p}:=\sum _j s^p_j(T)<\infty . \end{aligned}$$

The functional \(\Vert \cdot \Vert _{{\mathfrak {S}}_p}\) is the norm on \({\mathfrak {S}}_p\).

Let \(K\in {\mathfrak {S}}_n\) with \(n\in {\mathbb {N}}\). Let \(\lambda _j(K)\) denote the eigenvalues of K, repeated according to algebraic multiplicities. The n-th order regularized determinant \(\det {}_n(1+K)\) is defined by

$$\begin{aligned} \det {}_n(1+K) := \prod _j \left( \left( 1+\lambda _j(K) \right) \exp \left( \sum _{m=1}^{n-1} \frac{(-1)^m}{m} \lambda _j(K)^m \right) \right) \,. \end{aligned}$$

The following property is well-known, but we include a proof for the sake of completeness.

Lemma 3.1

Let \(n\in \mathbb {N}\) and let \(K\in {\mathfrak {S}}_n\). Then

$$\begin{aligned} \ln |\det {}_n(1+K)| \le \Gamma _{n} \Vert K\Vert _{{\mathfrak {S}}_n}^n \,, \end{aligned}$$

where \(\Gamma _{n}\) is a positive constant independent of K. In particular,

$$\begin{aligned} \Gamma _{1}=1 \quad and \quad \Gamma _{2} = 1/2 \,. \end{aligned}$$
(3.1)

Proof

To prove the lemma, let \(f(z) := (1+z) \exp \left( \sum _{m=1}^{n-1} \frac{(-1)^m}{m} z^m \right) \). Then \(\ln |f(z)|\) can be bounded by a constant times \(|z|^n\) for small |z| and by a constant times \(|z|^{n-1}\) for large |z|. Thus, \(\ln |f(z)|\le \Gamma _{n}|z|^n\), and so

$$\begin{aligned} \ln |\det {}_n(1+K)| \le \Gamma _{n} \sum _j |\lambda _j(K)|^n \end{aligned}$$

By Weyl’s inequality [26, Thm. 1.15], the sum on the right side does not exceed \(\Vert K\Vert _{{\mathfrak {S}}_n}^n\). A simple computation shows that for \(n=1\) and \(n=2\) one can take \(\Gamma _{1}=1\) and \(\Gamma _{2}=1/2\), respectively (see [27]). \(\square \)

We now recall the Birman–Schwinger principle. We state it in the case, where \(H_0\) is a bounded self-adjoint operator and \(V=G^*G_0 \). We will assume that \(G_0\) and G are compact operators. Now, set

$$\begin{aligned} H=H_0+V. \end{aligned}$$

The Birman–Schwinger principle states that \(z\in \rho (H_0)\) is an eigenvalue of H if and only if \(-1\) is an eigenvalue of the Birman–Schwinger operator \(G_0(H_0-z)^{-1} G^*\). Moreover, the corresponding geometric multiplicities coincide.

The following lemma says that even the algebraic multiplicities of eigenvalues of H can be characterizes in terms of a quantity related to the Birman–Schwinger operator.

Lemma 3.2

Let \(n\in \mathbb {N}\). Assume that \(G_0(H_0-\zeta )^{-1}G^*\in {\mathfrak {S}}_n\) for all \(\zeta \in \rho (H_0)\). Then the function \(\zeta \mapsto \det {}_n(1+G_0(H_0-\zeta )^{-1}G^*)\) is analytic in \(\rho (H_0)\). A point \(z\in \rho (H_0)\) is an eigenvalue of H if and only if \(\det {}_n(1+G_0(H_0-z)^{-1}G^*)=0\). Moreover, the order of the zero coincides with the algebraic multiplicity of the corresponding eigenvalue.

Analyticity of the function \(\zeta \mapsto \det {}_n(1+G_0(H_0-\zeta )^{-1}G^*)\) is well-known (see, e.g., [27]), as well as the result about the algebraic multiplicity in the case \(n=1\). The result for the general n is essentially due to [18]; you may also refer to [11] for an extension of the proof to the present setting.

4 Resolvent bounds

In this section we collect trace ideal bounds for the Birman–Schwinger operator

$$\begin{aligned} K(k) = \sqrt{V} (H_0 -z)^{-1} \sqrt{|V|}, \quad z=k+k^{-1},\,\,\,\, |k|\ge 1\,. \end{aligned}$$
(4.1)

We use the notation \(\sqrt{V(x)} = V(x)/\sqrt{|V(x)|}\) if \(V(x)\ne 0\) and \(\sqrt{V(x)}=0\) if \(V(x)=0\).

We remind the reader that \({\mathfrak {H}}=\ell ^2({\mathbb {Z}}_+)\), and \(H_0\) in (4.1) denotes the free Jacobi operator on \({\mathbb {Z}}_+\). From the explicit expression of its matrix it is easy to see that, if V has a compact support, then K(k) admits an analytic continuation to \(\mathbb {C}\setminus \{0\}\). The following proposition gives a bound on the Hilbert–Schmidt norm.

Proposition 4.1

For any \(k\in \mathbb {C}{\setminus }\{0\}\) with \(|k|<1\),

$$\begin{aligned} \Vert K(k) \Vert _{{\mathfrak {S}}_2} \le \frac{2}{1-|k|^2} \sum _{n=1}^\infty | k|^{-2n}|V_n| \,, \end{aligned}$$

Proof

The matrix of \((H_0 -z)^{-1}\) is given by

$$\begin{aligned} g_k(n,m) = \frac{k}{k^2-1} \left( k^{-|n-m|} - k^{-(n+m) }\right) \,, \end{aligned}$$

which satisfies

$$\begin{aligned} |g_k(n,m) | \le \frac{2}{1-|k|^2}| k|^{-(n+m)} \,. \end{aligned}$$

Combining this bound with the identity

$$\begin{aligned} \Vert K(k) \Vert _{{\mathfrak {S}}_2}^2 = \sum _1^\infty \sum _1^\infty |V_n| |g_k(n,m)|^2 |V_m| \end{aligned}$$

we obtain the claimed bound.

\(\square \)

Proposition 4.2

For any \(k\in \mathbb {C}{\setminus }\{0\}\) with \(|k|<1\),

$$\begin{aligned} \Vert K(k) \Vert _{{\mathfrak {S}}_1} \le \frac{2}{1-|k|^2}\Bigl ( \sum _{n=1}^\infty | k|^{-n}|V_n|^{1/2} \Bigr )^2\,, \end{aligned}$$

Proof

The matrix of \((H_0 -z)^{-1}\) is defined by

$$\begin{aligned} g_k(n,m) = \frac{k}{k^2-1} \left( k^{-|n-m|} - k^{-(n+m) }\right) \,, \end{aligned}$$

which satisfies

$$\begin{aligned} |g_k(n,m) | \le \frac{2}{1-|k|^2}| k|^{-(n+m)} \,. \end{aligned}$$

Combining this bound with the identity

$$\begin{aligned} \Vert K(k) \Vert _{{\mathfrak {S}}_1}\le \sum _1^\infty \sum _1^\infty |V_n|^{1/2} |g_k(n,m)| |V_m|^{1/2} \end{aligned}$$

we obtain the claimed bound. \(\square \)

5 Proof of Theorem 1.1

In this section we prove Theorem 1.1. Let us assume that V has compact support. The bound in this case implies the bound in the general case by a simple continuity argument.

As discussed in Sect. 4, the Birman–Schwinger operators K(k) from (4.1) extends analytically to \(\mathbb {C}{\setminus }\{0\}\). The same proof shows that they are not only analytic with respect to the norm of bounded operators, but even with respect to the norm in \({\mathfrak {S}}_{2}\).

We will apply Corollary 2.2 to the function

$$\begin{aligned} a(k) := \det {}_{2}(1+K(k)) \end{aligned}$$

with \(\Lambda = 1/R\). Since K(k) is analytic with values in \({\mathfrak {S}}_{2}\), the function a is analytic. It is easy to see that assumption (2.1) is valid. Moreover, combining them with Lemma 3.1, we see that assumption (2.2) holds with

$$\begin{aligned} \ln A = \frac{1}{2} \left( \frac{2\Lambda ^2}{\Lambda ^2-1}\sum _{n=1}^\infty \Lambda ^{2 n} |V_n| \right) ^2 \,. \end{aligned}$$

Thus, Corollary 2.2 implies that

$$\begin{aligned} \#\{ j:\,\ {{\mathrm{Im}}}k_j \ge 0\} \le \frac{1}{2\ln \Lambda } \left( \frac{2\Lambda ^2}{\Lambda ^2-1}\sum _{n=1}^\infty \Lambda ^{ 2n} |V_n| \right) ^2 \,. \end{aligned}$$

It remains to use Lemma 3.2, which says that the \(k_j+k_j^{-1}\), with \(| k_j|>1 \), coincide with the eigenvalues of H, counting algebraic multiplicities. This proves Theorem 1.1.

6 Proof of Theorem 1.2

In this section we prove Theorem 1.2. Let us assume again that V has compact support.

As discussed in Sect. 4, the Birman–Schwinger operators K(k) from (4.1) extend analytically to \(\mathbb {C}{\setminus }\{0\}\). The same proof shows that they are not only analytic with respect to the norm of bounded operators, but even with respect to the norm in \({\mathfrak {S}}_{1}\).

We apply Corollary 2.2 to the function

$$\begin{aligned} a(k) := \det {}_{1}(1+K(k))=\det {}(1+K(k)) \end{aligned}$$

with \(\Lambda = 1/R\). Since K(k) is analytic with values in \({\mathfrak {S}}_{1}\), the function a is analytic. Assumption (2.2) holds with

$$\begin{aligned} \ln A = \left( \frac{2\Lambda ^2}{\Lambda ^2-1}\sum _{n=1}^\infty \Lambda ^{n} |V_n|^{1/2} \right) ^2 \,. \end{aligned}$$

Thus, Corollary 2.2 implies that

$$\begin{aligned} \#\{ j:\,\ {{\mathrm{Im}}}k_j \ge 0\} \le \frac{1}{\ln \Lambda } \left( \frac{2\Lambda ^2}{\Lambda ^2-1}\sum _{n=1}^\infty \Lambda ^{ n} |V_n|^{1/2} \right) ^2 \,. \end{aligned}$$

It remains to use Lemma 3.2, which says that the \(k_j+k_j^{-1}\), with \(| k_j|>1 \), coincide with the eigenvalues of H, counting algebraic multiplicities. This proves Theorem 1.2. \(\square \)

Most of the papers listed below contain results on the eigenvalues of non-selfadjoint operators. More specifically, those are the articles [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17, 19,20,21,22,23,24,25, 28, 29]. The remaining references were needed for technical reasons.