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.
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1 Introduction
In this paper, we consider bounded non-selfadjoint Jacobi operators on \(\ell ^2(\mathbb Z)\) represented by tridiagonal matrices
with bounded complex parameters \(\{a_n,b_n,c_n\}_{n\in \mathbb Z}\). Our goal is to obtain Lieb–Thirring inequalities for complex perturbations of periodic and, more generally, almost periodic Jacobi operators with absolutely continuous finite gap spectrum.
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.
To put our new results in perspective, we first discuss the best currently known results on eigenvalue power bounds for Jacobi operators in more detail. The spectral theory for perturbations of the 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,
with some explicit constants \(L_{p,\,\mathsf {E}}\) independent of 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 \(\mathsf {E}\).
As alluded to above, there is a general result of Kato [24] which applies to compact selfadjoint perturbations of arbitrary bounded selfadjoint operators. Specialized to the case of perturbations of Jacobi matrices from \(\mathcal T_\mathsf {E}\), it states that
where \(\Vert {\cdot }\Vert _p\) denotes the Schatten norm. In contrast to the Lieb–Thirring bounds, the power on the eigenvalues in (1.3) is the same as on the perturbation and so is larger than the power on the eigenvalues in (1.2) by 1 / 2. Kato’s inequality is optimal for perturbations with large sup norm. On the other hand, the Lieb–Thirring bound with \(p=1\) is optimal for perturbations with small sup norm (cf. [21]). A fact that seemingly went unnoticed is that one can combine (1.2) and (1.3) into one ultimate inequality which is optimal for both large and small perturbations (at least when \(p=1\)). This inequality takes the form
where the constant \(C_{p,\,\mathsf {E}}\) is independent of 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.
In recent years, several results have also been established for non-selfadjoint perturbations of selfadjoint Jacobi matrices [1, 16,17,18,19]. For compact non-selfadjoint perturbations \(J=J_0+\delta J\) of the free Jacobi matrix \(J_0\), a near generalization (with an extra \(\varepsilon \)) of the Lieb–Thirring bound (1.2) was obtained by Hansmann and Katriel [19] using the complex analytic approach developed in [1]. Their non-selfadjoint version of the Lieb–Thirring inequalities takes the following form: 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.
For non-selfadjoint perturbations of Jacobi matrices from finite gap isospectral tori \(\mathcal T_\mathsf {E}\), an eigenvalue power bound was first obtained by Golinskii and Kupin in [16]. Shortly thereafter, this bound was superseded by a generalization of Kato’s inequality to non-selfadjoint perturbations of arbitrary bounded selfadjoint operators (see Hansmann [18]). In the special case of a compact non-selfadjoint perturbation \(J=J'+\delta J\) of \(J'\in \mathcal T_\mathsf {E}\), Hansmann’s result reads
where the eigenvalues are repeated according to their algebraic multiplicity and \(K_p\) is a universal constant that depends only on p.
The purpose of the present article is to generalize the Lieb–Thirring bound (1.4) to the case of compact non-selfadjoint perturbations \(J=J'+\delta J\) of Jacobi matrices \(J'\) from finite gap isospectral tori \(\mathcal T_\mathsf {E}\). Let \(\partial \mathsf {E}\) denote the set of endpoints of the intervals that form \(\mathsf {E}\). Then, our main result can be formulated as follows: 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
Suppose \(J'\) is a selfadjoint Jacobi matrix and \(D\ge 0\) is a diagonal matrix of Schatten class \(\mathcal S_p\) for some \(p\ge 1\). Denote by \(\hbox {d}\rho _n\) the spectral measure of \(\left( J',\delta _n\right) \), that is, the measure in the Herglotz representation of the nth diagonal entry of \(\left( J'-z\right) ^{-1}\),
Then,
for .
Proof
We consider first the case \(p=1\). Let \(\{P(t)\}_{t\in \mathbb R}\) denote the projection-valued spectral family of the selfadjoint operator \(J'\). Recall that for any measurable and bounded function f on \(\sigma \left( J'\right) \), the bounded operator \(f\left( J'\right) \) is given by the functional calculus,
Taking \(f(t)=1/(t-z)\) in (2.3), substituting into (2.1), and recalling that the measure in the Herglotz representation is unique yield
Applying (2.3) to \(f(t)=1/|t-z|\) and using (2.4) then imply
We also note that if, in addition, the function f(t) in (2.3) is nonnegative, then \(f\left( J'\right) \) is a bounded, selfadjoint, and nonnegative operator.
Fix . In the following, we assume without loss of generality that \(\mathrm{Im}(z)\ge 0\). Define the nonnegative functions
and note that
Then, we have \(f\left( J'\right) \ge 0\), \(f_\pm \left( J'\right) \ge 0\), and
Using (2.9), the triangle inequality, and the fact that for nonnegative operators the trace norm coincides with the trace, we obtain the estimate
Let \(D_{n,n}\) denote the diagonal entries of D. Since D is nonnegative and diagonal, we have
Hence, by linearity of the trace it follows from (2.11), (2.10), and (2.5) that
This is exactly the case \(p=1\) in (2.2).
To obtain (2.2) for \(p>1\), we use complex interpolation. Define the map
from the strip \(0\le \mathrm{Re}(\zeta )\le 1\) into the space of bounded operators. Then, for any \(u,v\in \ell ^2(\mathbb Z)\), the scalar function
is continuous on the strip \(0\le \mathrm{Re}(\zeta )\le 1\), analytic in its interior, and bounded. In addition, since \(\Vert D^{iy}\Vert \le 1\) and
it follows that
and by [28, Theorem 2.7] and (2.13),
Thus, by the complex interpolation theorem (see [28, Theorem 2.9], [14, Theorem III.13.1]), we have
Taking \(x=1/p\), raising both sides to the power 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). \(\square \)
In what follows, \(\mathsf {E}\subset \mathbb R\) will denote a finite gap set, that is,
and \(\partial \mathsf {E}\) will be the set of endpoints of \(\mathsf {E}\), that is,
For a probability measure \(\hbox {d}\rho \) supported on \(\mathsf {E}\), we define the associated m-function by
The measure \(\hbox {d}\rho \) is called reflectionless (on \(\mathsf {E}\)) if
Reflectionless measures appear prominently in spectral theory of finite and infinite gap Jacobi matrices (see, e.g., [2, 27, 29, 30]). In particular, the isospectral torus associated with \(\mathsf {E}\) is the set of all Jacobi matrices \(J'\) that are reflectionless on \(\mathsf {E}\) (i.e., the spectral measure of \(\left( J',\delta _n\right) \) is reflectionless for every \(n\in \mathbb Z\)) and for which \(\sigma \left( J'\right) =\mathsf {E}\). It is well known (see for example [30]) that \(\hbox {d}\rho \) is a reflectionless probability measure on \(\mathsf {E}\) if and only if m(z) is of the form
for some \(\gamma _j\in [\beta _j,\alpha _{j+1}]\), \(j=1,\dots ,N-1\).
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
Let \(\mathsf {E}\subset \mathbb R\) be a finite gap set and suppose \(\hbox {d}\rho \) is a reflectionless probability measure on \(\mathsf {E}\). Then, for every \(p>1\),
where the constant \(K_{p,\,\mathsf {E}}\) is independent of \(\hbox {d}\rho \). In addition, for every \(\varepsilon >0\),
where the constant \(K_{\varepsilon ,\,\mathsf {E}}\) is independent of \(\hbox {d}\rho \).
Proof
Denote the bands of \(\mathsf {E}\) as in (2.20). Since \(\hbox {d}\rho \) is reflectionless on a finite gap set, it follows from the Stieltjes inversion formula and (2.24) that \(\hbox {d}\rho \) is absolutely continuous with density given by
for some \(\gamma _j\in [\beta _j,\alpha _{j+1}]\), \(j=1,\dots ,N-1\). Fix \(1\le k\le N\) and rearrange the terms in (2.27) as follows
Since \(\alpha _j<\beta _j\le \gamma _j\le \alpha _{j+1}<\beta _{j+1}\) for \(j=1,\dots ,N-1\), the two products in (2.28) are at most 1 for every \(t\in [\alpha _k,\beta _k]\), and thus,
Applying this estimate for the individual bands of \(\mathsf {E}\) implies that
By [19, Lemma 11], for \(p>1\), each integral in the sum can be estimated by
Since the function \(z\mapsto {\mathrm{dist}}(z,\partial \mathsf {E})(1+|z|)/|z-\alpha _k||z-\beta _k|\) is continuous on and bounded near \(\alpha _k\), \(\beta _k\), and \(\infty \), it is bounded on , and therefore,
Combining (2.32) with (2.30) yields (2.25).
In order to obtain (2.26), note that since \(\mathsf {E}\) is a bounded set we have the trivial bound
This inequality yields the estimate
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
Let \(S\subset \partial \mathbb D\) be a finite collection of points and suppose h(z) is an analytic function on \(\mathbb D\) such that \(|h(0)|=1\) and for some \(K,\alpha ,\beta ,\gamma \ge 0\),
Then, for every \(\varepsilon >0\), there exists a constant \(C_{\alpha ,\beta ,\gamma ,\varepsilon }\) independent of h(z) such that the zeros of h(z) satisfy
where each zero is repeated according to its multiplicity.
In [16], an analogous result on the distribution of zeros of analytic functions on 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
Let \(\mathsf {E}\subset \mathbb R\) be a finite gap set and suppose f(z) is an analytic function on such that \(|f(\infty )|=1\) and for some \(K,p,q,r\ge 0\),
Then, for every \(\varepsilon >0\), there exists a constant \(C_{p,q,r,\varepsilon }\) independent of f(z) such that the zeros of f(z) satisfy
where \(p' = p+1+\varepsilon \), \(q' = \frac{1}{2}[(p+2q-1+\varepsilon )_+ - p']\), \(r' = (p+q+r-\varepsilon )_+ - p' - q'\), and each zero is repeated according to its multiplicity.
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
Let \(\mathsf {E}\subset \mathbb R\) be a finite gap set and suppose 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 \(\varepsilon >0\),
where the eigenvalues are repeated according to their algebraic multiplicity and the constant \(L_{\varepsilon ,\,p,\,\mathsf {E}}\) is independent of J and \(J'\).
Proof
Suppose that \(\delta J\in \mathcal S_p\) for some \(p\ge 1\) and define \(D\ge 0\) to be the diagonal matrix with the entries
A straightforward verification shows that \(\delta J = D^{1/2}BD^{1/2}\), where B is a bounded tridiagonal matrix whose entries lie in the unit disk. This in particular means that \(\Vert B\Vert \le 3\). Define
where \(\lceil p\rceil \) is the smallest integer \(\ge p\). This regularized perturbation determinant is analytic on (see, e.g., [14, Chapter IV. Sect. 3]). More importantly, the zeros of 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
for some constant \(K_p\). It thus follows from Theorems 2.1 and 2.2 (with \(\varepsilon /2\) instead of \(\varepsilon \)) that
Applying Theorem 3.2 (with \(\varepsilon /2\) instead of \(\varepsilon \)) and noting that \({(1-3\varepsilon )/2}\le r'\) and \(D_{n,n}^p\le |\delta a_{n-1}|^p+|\delta a_n|^p+|\delta b_n|^p+|\delta c_{n-1}|^p+|\delta c_n|^p\) then yield (3.5). \(\square \)
References
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)
Christiansen, J.S.: Dynamics in the Szegő class and polynomial asymptotics. J. Anal. Math. (to appear)
Christiansen, J.S., Simon, B., Zinchenko, M.: Finite gap Jacobi matrices, I. The isospectral torus. Constr. Approx. 32, 1–65 (2010)
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
Damanik, D., Killip, R., Simon, B.: Perturbations of orthogonal polynomials with periodic recursion coefficients. Ann. Math. 171, 1931–2010 (2010)
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.: 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)
Dunford, N., Schwartz, J.T.: Linear operators. Part II: spectral theory, self adjoint operators in Hilbert space. Wiley, New York (1963)
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)
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)
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)
Frank, R.L., Sabin, J.: Restriction theorems for orthonormal functions, Strichartz inequalities, and uniform Sobolev estimates. Am. J. Math. (to appear)
Frank, R.L.: Eigenvalue bounds for Schrödinger operators with complex potentials. III. Trans. Am. Math. Soc. (to appear)
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)
Golinskii, L., Kupin, S.: Lieb–Thirring bounds for complex Jacobi matrices. Lett. Math. Phys. 82(1), 79–90 (2007)
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)
Hansmann, M.: An eigenvalue estimate and its application to non-selfadjoint Jacobi and Schrödinger operators. Lett. Math. Phys. 98(1), 79–95 (2011)
Hansmann, M.: Variation of discrete spectra for non-selfadjoint perturbations of selfadjoint operators. Integral Equ. Oper. Theory 76(2), 163–178 (2013)
Hansmann, M., Katriel, G.: Inequalities for the eigenvalues of non-selfadjoint Jacobi operators. Complex Anal. Oper. Theory 5(1), 197–218 (2011)
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)
Hundertmark, D., Simon, B.: Lieb–Thirring inequalities for Jacobi matrices. J. Approx. Theory 118, 106–130 (2002)
Hundertmark, D., Simon, B.: Eigenvalue bounds in the gaps of Schrödinger operators and Jacobi matrices. J. Math. Anal. Appl. 340(2), 892–900 (2008)
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)
Kato, T.: Variation of discrete spectra. Commun. Math. Phys. 111(3), 501–504 (1987)
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)
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)
Remling, C.: The absolutely continuous spectrum of Jacobi matrices. Ann. Math. (2) 174(1), 125–171 (2011)
Simon, B.: Trace Ideals and Their Applications, Mathematical Surveys and Monographs, vol. 120, pp. viii+150. American Mathematical Society, Providence (2005)
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)
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)
Weidl, T.: On the Lieb–Thirring constants \(L_{\gamma,1}\) for \(\gamma \ge 1/2\). Commun. Math. Phys. 178, 135–146 (1996)
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JSC is supported in part by the Research Project Grant DFF-4181-00502 from the Danish Council for Independent Research. MZ is supported in part by Simons Foundation Grant CGM-281971.
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Christiansen, J.S., Zinchenko, M. Lieb–Thirring inequalities for complex finite gap Jacobi matrices. Lett Math Phys 107, 1769–1780 (2017). https://doi.org/10.1007/s11005-017-0961-z
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DOI: https://doi.org/10.1007/s11005-017-0961-z