Abstract
We study the eigenvalues of the discrete Schrödinger operator with a complex potential. We obtain bounds on the total number of eigenvalues in the case where V decays exponentially at infinity.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
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
Additionally, we set
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
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
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
Assume also that for some \(A\ge 1\),
Then the zeroes \(k_j\) of \(a(\cdot )\) in \(\{k:\,|k|>R\}\), repeated according to their multiplicities, satisfy
Proof
We introduce the Blaschke product
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,
and therefore
We note that \(|B(Re^{i\varphi })|=1\) if \(\varphi \in \mathbb {R}\) and, therefore,
On the other hand,
We conclude from (2.4), (2.5) and (2.6) that
Finally, by (2.2),
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
of zeroes \(k_j\) of \(a(\cdot )\) in \(\{ k:\, |k|\ge 1\}\), repeated according to their multiplicities, satisfies
Proof
We apply Proposition 2.1 for every \(R'>R\) sufficiently close to R and obtain
Clearly, we have
Consequently,
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
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
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
where \(\Gamma _{n}\) is a positive constant independent of K. In particular,
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
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
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
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 \(|k|<1\),
Proof
The matrix of \((H_0 -z)^{-1}\) is given by
which satisfies
Combining this bound with the identity
we obtain the claimed bound.
\(\square \)
Proposition 4.2
For any \(k\in \mathbb {C}{\setminus }\{0\}\) with \(|k|<1\),
Proof
The matrix of \((H_0 -z)^{-1}\) is defined by
which satisfies
Combining this bound with the identity
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
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
Thus, Corollary 2.2 implies that
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
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
Thus, Corollary 2.2 implies that
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.
References
Abramov, A.A., Aslanyan, A., Davies, E.B.: Bounds on complex eigenvalues and resonances. J. Phys. A 34, 57–72 (2001)
Bögli, S.: Schrödinger operator with non-zero accumulation points of complex Eigenvalues (2016). arxiv:1605.09356 (Preprint)
Borichev, A., Golinskii, L., Kupin, S.: A Blaschke-type condition and its application to complex Jacobi matrices. Bull. Lond. Math. Soc. 41, 117–123 (2009)
Davies, E.B.: Non-self-adjoint differential operators. Bull. Lond. Math. Soc. 34(5), 513–532 (2002)
Davies, E.B., Nath, J.: Schrödinger operators with slowly decaying potentials. J. Comput. Appl. Math. 148, 1–28 (2002)
Demuth, M., Katriel, G.: Eigenvalue inequalities in terms of Schatten norm bounds on differences of semigroups, and application to Schrödinger operators. Ann. Henri Poincaré 9(4), 817–834 (2008)
Demuth, M., Hansmann, M., Katriel, G.: On the discrete spectrum of non-selfadjoint operators. J. Funct. Anal. 257(9), 2742–2759 (2009)
Demuth, M., Hansmann, M., Katriel, G.: Eigen values of non-self adjoint operators: a comparison of two approaches. In: Mathematical Physics, Spectral Theory and Stochastic Analysis, Springer, pp. 107–163 (2013)
Enblom, A.: Estimates for eigenvalues of Schrödinger operators with complex-valued potentials (2015). arxiv:1503.06337 (Preprint)
Frank, R.L.: Eigenvalue bounds for Schrödinger operators with complex potentials. Bull. Lond. Math. Soc. 43(4), 745–750 (2011)
Frank, R.L.: Eigenvalue bounds for Schrödinger operators with complex potentials. III (2015). arxiv:1510.03411v1 (Preprint)
Frank, R.L., Laptev, A., Lieb, E.H., Seiringer, R.: Lieb-Thirring inequalities for Schrödinger operators with complex-valued potentials. Lett. Math. Phys. 77, 309–316 (2006)
Frank, R.L., Laptev, A., Safronov, O.: On the number of eigenvalues of Schrödinger operators with complex potentials. to appear
Frank, R.L., Laptev, A., Seiringer, R.: A sharp bound on eigenvalues of Schrödinger operators on the half-line with complex-valued potentials. Spectral theory and analysis, pp. 39–44, Oper. Theory Adv. Appl. 214, Birkhäuser/Springer Basel AG, Basel (2011)
Frank, R.L., Sabin, J.: Restriction theorems for orthonormal functions, Strichartz inequalities and uniform Sobolev estimates (2014). arxiv:1404.2817 (Preprint)
Frank, R.L., Simon, B.: Eigenvalue bounds for Schrödinger operators with complex potentials. II. J. Spectr. Theory (to appear)
Laptev, A., Safronov, O.: Eigenvalue estimates for Schrödinger operators with complex potentials. Comm. Math. Phys. 292, 29–54 (2009)
Latushkin, Y., Sukhtayev, A.: The algebraic multiplicity of eigenvalues and the Evans function revisited. Math. Model. Nat. Phenom. 5(4), 269–292 (2010)
Martirosjan, R.M.: On the spectrum of the non-selfadjoint operator \(\Delta u+cu\) in three dimensional space. (Russian) Izv. Akad. Nauk Armyan. SSR. Ser. Fiz.-Mat. Nauk 10(1), 85–111 (1957)
Martirosjan, R.M.: On the spectrum of various perturbations of the Laplace operator in spaces of three or more dimensions. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 24, 897–920 (1960)
Murtazin, K.K.: Spectrum of the nonself-adjoint Schrödinger operator in unbounded regions. (Russian) Mat. Zametki 9, 19–26. English translation: Math. Notes 9(1971), 12–16 (1971)
Naĭmark, M.A.: Investigation of the spectrum and the expansion in eigenfunctions of a nonselfadjoint operator of the second order on a semi-axis. (Russian) Trudy Moskov. Mat. Obšč. 3, 181–270 (1954)
Pavlov, B.S.: On a non-selfadjoint Schrödinger operator. (Russian) 1966 Probl. Math. Phys., No. 1, Spectral Theory and Wave Processes (Russian) pp. 102–132 Izdat. Leningrad. Univ., Leningrad
Pavlov, B.S.: On a non-selfadjoint Schrödinger operator. II. (Russian) 1967 Problems of Mathematical Physics, No. 2, Spectral Theory, Diffraction Problems (Russian) pp. 133–157 Izdat. Leningrad. Univ., Leningrad
Safronov, O.: On a sum rule for Schrödinger operators with complex potentials. Proc. Am. Math. Soc. 138(6), 2107–2112 (2010)
Simon, B.: Trace Ideals and Their Applications, 2nd edn. Amer. Math. Soc, Providence (2005)
Simon, B.: Notes on infinite determinants of Hilbert space operators. Adv. Math. 24(3), 244–273 (1977)
Stepin, S.A.: Complex potentials: bound states, quantum dynamics and wave operators. Semigroups of operators - theory and applications, 287–297, Springer Proc. Math. Stat. 113, Springer, Cham (2015)
Stepin, S.A.: Estimate for the number of eigenvalues of the nonselfadjoint Schrödinger operator. (Russian) Dokl. Akad. Nauk 455(4), 394–397; translation in Dokl. Math. 89(2), 202–205 (2014)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Ari Laptev.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Hulko, A. On the number of eigenvalues of the discrete one-dimensional Schrödinger operator with a complex potential. Bull. Math. Sci. 7, 219–227 (2017). https://doi.org/10.1007/s13373-016-0093-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13373-016-0093-2