# Lieb–Thirring inequalities for complex finite gap Jacobi matrices

## Abstract

We establish Lieb–Thirring power bounds on discrete eigenvalues of Jacobi operators for Schatten class complex perturbations of periodic and more generally finite gap almost periodic Jacobi matrices.

## Keywords

Finite gap Jacobi matrices Complex perturbations Eigenvalues estimates## Mathematics Subject Classification

34L15 47B36## 1 Introduction

Lieb–Thirring inequalities for selfadjoint and complex perturbations of the discrete Laplacian have been studied extensively in the last decade [1, 7, 15, 17, 19, 21]. The original work of Lieb and Thirring [25, 26] was carried out in the context of continuous Schrödinger operators, motivated by their study of the stability of matter. We refer to [6, 9, 10, 12, 13, 23, 31] for more recent developments on Lieb–Thirring inequalities for Schrödinger operators and to [20] for a review and history of the subject.

Much less is known for perturbations (especially complex ones) of operators with gapped spectrum. Lieb–Thirring inequalities for selfadjoint perturbations of periodic and almost periodic Jacobi operators with absolutely continuous finite gap spectrum have been established only recently [4, 5, 11, 22]. Analogs of these finite gap Lieb–Thirring inequalities for complex perturbations are not known. The aim of the present work is to fill this gap. What is currently known in the case of complex perturbations is the closely related class of Kato inequalities [16, 18]. Such inequalities have larger exponents on the eigenvalue side when compared to Lieb–Thirring inequalities [cf. (1.3) vs. (1.2) and (1.6) vs. (1.7)] and hence are not optimal for small perturbations of Jacobi operators.

*free*Jacobi matrix, \(J_0\), (i.e., the case of \(a_n=c_n\equiv 1\) and \(b_n\equiv 0\)) is well understood, see [29]. Let \(\mathsf {E}=\sigma (J_0)=[-2,2]\) and suppose

*J*is a selfadjoint Jacobi matrix (i.e., \(a_n=c_n>0\)) such that \(\delta J=J-J_0\) is a compact operator, that is,

*J*is a compact selfadjoint perturbation of \(J_0\). Hundertmark and Simon [21] proved the following Lieb–Thirring inequalities,

*J*. Here, \(\sigma _{d}(J)\) is the discrete spectrum of

*J*. It was also shown in [21] that the inequality is false for \(p<1\).

More recently, (1.2) was extended to selfadjoint perturbations of periodic and almost periodic Jacobi matrices with absolutely continuous finite gap spectrum [4, 5, 11, 22]. When \(\mathsf {E}\) is a finite gap set (i.e., a finite union of disjoint, compact intervals), the role of \(J_0\) as a natural background operator is taken over by the so-called isospectral torus, denoted \(\mathcal T_\mathsf {E}\). See, e.g., [2, 3, 27, 29, 30] for a deeper discussion of this object. For \(J'\in \mathcal T_\mathsf {E}\) and a compact selfadjoint perturbation \(J=J'+\delta J\), Frank and Simon [11] proved (1.2) for \(p=1\) while the case of \(p>1\) is established in [4]. The constant \(L_{p,\,\mathsf {E}}\) is now independent of *J* and \(J'\) and only depends on *p* and the underlying set \(\mathsf {E}\).

*J*and \(J'\), \(J=J'+\delta J\), \(\delta J\) is compact, \(J'\in \mathcal T_\mathsf {E}\), and \(\mathsf {E}\) is a finite gap set.

*For every*\(0<\varepsilon <1\),

*where the eigenvalues are repeated according to their algebraic multiplicity and the constant*\(L_{p,\,\varepsilon }\)

*is independent of*

*J*. Whether or not this inequality continues to hold for \(\varepsilon =0\) is an open problem.

*p*.

*For every*\(p\ge 1\)

*and any*\(\varepsilon >0\),

*where the eigenvalues are repeated according to their algebraic multiplicity and the constant*\(L_{\varepsilon ,\,p,\,\mathsf {E}}\)

*is independent of*\(J'\) and

*J*. We note that for the eigenvalues that accumulate to \(\partial \mathsf {E}\), the inequality (1.7) gives a qualitatively better estimate than (1.6). We also point out that (1.7) is new even for perturbations of the free Jacobi matrix \(J_0\) since, unlike (1.5), it is nearly optimal not only for small but also for large perturbations. As with (1.5), it is an open problem whether or not (1.7) remains true for \(\varepsilon =0\).

## 2 Schatten norm estimates

In this section, we establish the fundamental estimates that are needed to prove our main result, Theorem 3.3. Throughout, \(\mathcal S_p\) will denote the Schatten class and \(\Vert {\cdot }\Vert _p\) the corresponding Schatten norm for \(p\ge 1\). To clarify our application of complex interpolation, we occasionally use \(\Vert {\cdot }\Vert _\infty \) to denote the operator norm.

### Theorem 2.1

*n*th diagonal entry of \(\left( J'-z\right) ^{-1}\),

### Proof

*f*on \(\sigma \left( J'\right) \), the bounded operator \(f\left( J'\right) \) is given by the functional calculus,

*f*(

*t*) in (2.3) is nonnegative, then \(f\left( J'\right) \) is a bounded, selfadjoint, and nonnegative operator.

*D*. Since

*D*is nonnegative and diagonal, we have

*p*, and noting that \(T(1/p)=D^{1/2}\left( J'-z\right) ^{-1}D^{1/2}\) and \(\Vert D^p\Vert _1=\Vert D\Vert _p^p\) finally yield (2.2). \(\square \)

*m*-function by

*reflectionless*(on \(\mathsf {E}\)) if

*m*(

*z*) is of the form

We now provide an estimate for the variant of the *m*-function for \(\hbox {d}\rho _n\) that appear in Theorem 2.1.

### Theorem 2.2

### Proof

## 3 Lieb–Thirring bounds

We start this section by recalling some results on the distribution of zeros of analytic functions with restricted growth toward the boundary of the domain of analyticity. Let \(a_+\) denote the maximum of *a* and 0. The following theorem for analytic functions on the unit disk is an alternative form of the extension [19, Theorem 4] of the earlier result [1, Theorem 0.2].

### Theorem 3.1

*h*(

*z*) is an analytic function on \(\mathbb D\) such that \(|h(0)|=1\) and for some \(K,\alpha ,\beta ,\gamma \ge 0\),

*h*(

*z*) such that the zeros of

*h*(

*z*) satisfy

In [16], an analogous result on the distribution of zeros of analytic functions on Open image in new window was obtained via a reduction to the unit disk case. For our purposes, we will need the following extension of [16, Theorem 0.1] where an additional decay assumption at infinity is imposed in exchange for a stronger conclusion. The extension follows from the reduction to the unit disk case developed in [16] combined with the above version (Theorem 3.1) of the unit disk result. We omit the proof as it is a straightforward modification of the one presented in [16].

### Theorem 3.2

*f*(

*z*) is an analytic function on Open image in new window such that \(|f(\infty )|=1\) and for some \(K,p,q,r\ge 0\),

*f*(

*z*) such that the zeros of

*f*(

*z*) satisfy

We are now ready to present our finite gap version of the Lieb–Thirring inequalities for non-selfadjoint perturbations of Jacobi matrices from the isospectral torus \(\mathcal T_\mathsf {E}\).

### Theorem 3.3

*J*, \(J'\) are two-sided Jacobi matrices such that \(J'\in \mathcal T_\mathsf {E}\) and \(J=J'+\delta J\) is a compact perturbation of \(J'\). Then, for every \(p\ge 1\) and any \(\varepsilon >0\),

*J*and \(J'\).

### Proof

*B*is a bounded tridiagonal matrix whose entries lie in the unit disk. This in particular means that \(\Vert B\Vert \le 3\). Define

*f*coincide with the discrete eigenvalues of

*J*and the multiplicity of the zeros matches the algebraic multiplicity of the corresponding eigenvalues (see [19] and [13, Appendix C] for a proof). By [8, Lemma XI.9.22(d)], we have

## References

- 1.Borichev, A., Golinskii, L., Kupin, S.: A Blaschke-type condition and its application to complex Jacobi matrices. Bull. Lond. Math. Soc.
**41**(1), 117–123 (2009)MathSciNetCrossRefzbMATHGoogle Scholar - 2.Christiansen, J.S.: Dynamics in the Szegő class and polynomial asymptotics. J. Anal. Math. (
**to appear**)Google Scholar - 3.Christiansen, J.S., Simon, B., Zinchenko, M.: Finite gap Jacobi matrices, I. The isospectral torus. Constr. Approx.
**32**, 1–65 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - 4.Christiansen, J.S., Zinchenko, M.: Lieb-Thirring inequalities for finite and infinite gap Jacobi matrices. Ann. Henri Poincaré
**18**(6), 1949–1976 (2017). doi: 10.1007/s00023-016-0546-x MathSciNetCrossRefzbMATHGoogle Scholar - 5.Damanik, D., Killip, R., Simon, B.: Perturbations of orthogonal polynomials with periodic recursion coefficients. Ann. Math.
**171**, 1931–2010 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - 6.Demuth, M., Hansmann, M., Katriel, G.: On the discrete spectrum of non-selfadjoint operators. J. Funct. Anal.
**257**(9), 2742–2759 (2009)MathSciNetCrossRefzbMATHGoogle Scholar - 7.Demuth, M., Hansmann, M., Katriel, G.: Eigenvalues of non-selfadjoint operators: a comparison of two approaches. In: Mathematical Physics, Spectral Theory and Stochastic Analysis, Oper. Theory Adv. Appl., vol. 232, pp. 107–163. Birkhäuser, Basel (2013)Google Scholar
- 8.Dunford, N., Schwartz, J.T.: Linear operators. Part II: spectral theory, self adjoint operators in Hilbert space. Wiley, New York (1963)zbMATHGoogle Scholar
- 9.Frank, R.L., Laptev, A., Lieb, E.H., Seiringer, R.: Lieb–Thirring inequalities for Schrödinger operators with complex-valued potentials. Lett. Math. Phys.
**77**(3), 309–316 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 10.Frank, R.L., Simon, B., Weidl, T.: Eigenvalue bounds for perturbations of Schrödinger operators and Jacobi matrices with regular ground states. Commun. Math. Phys.
**282**, 199–208 (2008)ADSCrossRefzbMATHGoogle Scholar - 11.Frank, R.L., Simon, B.: Critical Lieb–Thirring bounds in gaps and the generalized Nevai conjecture for finite gap Jacobi matrices. Duke Math. J.
**157**(3), 461–493 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 12.Frank, R.L., Sabin, J.: Restriction theorems for orthonormal functions, Strichartz inequalities, and uniform Sobolev estimates. Am. J. Math. (
**to appear**)Google Scholar - 13.Frank, R.L.: Eigenvalue bounds for Schrödinger operators with complex potentials. III. Trans. Am. Math. Soc. (
**to appear**)Google Scholar - 14.Gohberg, I.C., Krein, M.G.: Introduction to the theory of linear nonselfadjoint operators. In: Translations of Mathematical Monographs, vol. 18, pp. xv+378. American Mathematical Society, Providence (1969)Google Scholar
- 15.Golinskii, L., Kupin, S.: Lieb–Thirring bounds for complex Jacobi matrices. Lett. Math. Phys.
**82**(1), 79–90 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 16.Golinskii, L., Kupin, S.: A Blaschke-type condition for analytic functions on finitely connected domains. Applications to complex perturbations of a finite-band selfadjoint operator. J. Math. Anal. Appl.
**389**(2), 705–712 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - 17.Hansmann, M.: An eigenvalue estimate and its application to non-selfadjoint Jacobi and Schrödinger operators. Lett. Math. Phys.
**98**(1), 79–95 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 18.Hansmann, M.: Variation of discrete spectra for non-selfadjoint perturbations of selfadjoint operators. Integral Equ. Oper. Theory
**76**(2), 163–178 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 19.Hansmann, M., Katriel, G.: Inequalities for the eigenvalues of non-selfadjoint Jacobi operators. Complex Anal. Oper. Theory
**5**(1), 197–218 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 20.Hundertmark, D.: Some bound state problems in quantum mechanics. In: Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simons 60th Birthday, Proc. Sympos. Pure Math., vol. 76, Part 1, pp. 463–496. Amer. Math. Soc., Providence (2007)Google Scholar
- 21.Hundertmark, D., Simon, B.: Lieb–Thirring inequalities for Jacobi matrices. J. Approx. Theory
**118**, 106–130 (2002)MathSciNetCrossRefzbMATHGoogle Scholar - 22.Hundertmark, D., Simon, B.: Eigenvalue bounds in the gaps of Schrödinger operators and Jacobi matrices. J. Math. Anal. Appl.
**340**(2), 892–900 (2008)MathSciNetCrossRefzbMATHGoogle Scholar - 23.Hundertmark, D., Lieb, E.H., Thomas, L.E.: A sharp bound for an eigenvalue moment of the one-dimensional Schrödinger operator. Adv. Theor. Math. Phys.
**2**, 719–731 (1998)MathSciNetCrossRefzbMATHGoogle Scholar - 24.Kato, T.: Variation of discrete spectra. Commun. Math. Phys.
**111**(3), 501–504 (1987)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 25.Lieb, E.H., Thirring, W.: Bound for the kinetic energy of fermions which proves the stability of matter. Phys. Rev. Lett.
**35**, 687–689 (1975). (Phys. Rev. Lett. 35 (1975) 1116, Erratum)ADSCrossRefGoogle Scholar - 26.Lieb, E.H., Thirring, W.: Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities. In: Studies in Mathematical Physics. Essays in Honor of Valentine Bargmann, pp. 269–303. Princeton University Press, Princeton (1976)Google Scholar
- 27.Remling, C.: The absolutely continuous spectrum of Jacobi matrices. Ann. Math. (2)
**174**(1), 125–171 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 28.Simon, B.: Trace Ideals and Their Applications, Mathematical Surveys and Monographs, vol. 120, pp. viii+150. American Mathematical Society, Providence (2005)Google Scholar
- 29.Simon, B.: Szegő’s Theorem and Its Descendants. Spectral Theory for \(L^2\) Perturbations of Orthogonal Polynomials, M. B. Porter Lectures, pp. xii+650. Princeton University Press, Princeton (2011)Google Scholar
- 30.Sodin, M., Yuditskii, P.: Almost periodic Jacobi matrices with homogeneous spectrum, infinite-dimensional Jacobi inversion, and Hardy spaces of character-automorphic functions. J. Geom. Anal.
**7**(3), 387–435 (1997)MathSciNetCrossRefzbMATHGoogle Scholar - 31.Weidl, T.: On the Lieb–Thirring constants \(L_{\gamma,1}\) for \(\gamma \ge 1/2\). Commun. Math. Phys.
**178**, 135–146 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar

## Copyright information

**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.