An official website of the United States government
From the general inverse theory of periodic Jacobi matrices, it is known that a periodic Jacobi matrix of minimal period p≥2 may have at most p−2 closed spectral gaps. We discuss the maximal number of closed gaps for one-dimensional periodic discrete Schrödinger operators of period p. We prove nontrivial upper and lower bounds on this quantity for large p and compute it exactly for p≤6. Among our results, we show that a discrete Schrödinger operator of period four or five may have at most a single closed gap, and we characterize exactly which potentials may exhibit a closed gap. For period six, we show that at most two gaps may close. In all cases in which the maximal number of closed gaps is computed, it is seen to be strictly smaller than p−2, the bound guaranteed by the inverse theory. We also discuss similar results for purely off-diagonal Jacobi matrices.
more »
« less
On a Conjecture by Hundertmark and Simon
Abstract The main result of this paper is a complete proof of a new Lieb–Thirring-type inequality for Jacobi matrices originally conjectured by Hundertmark and Simon. In particular, it is proved that the estimate on the sum of eigenvalues does not depend on the off-diagonal terms as long as they are smaller than their asymptotic value. An interesting feature of the proof is that it employs a technique originally used by Hundertmark–Laptev–Weidl concerning sums of singular values for compact operators. This technique seems to be novel in the context of Jacobi matrices.
more »
« less
- Award ID(s):
- 1856645
- PAR ID:
- 10367139
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Annales Henri Poincaré
- Volume:
- 23
- Issue:
- 11
- ISSN:
- 1424-0637
- Page Range / eLocation ID:
- p. 4057-4067
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
There is a vast theory of the asymptotic behavior of orthogonal polynomials with respect to a measure on R \mathbb {R} and its applications to Jacobi matrices. That theory has an obvious affine invariance and a very special role for ∞ \infty . We extend aspects of this theory in the setting of rational functions with poles on R ¯ = R ∪ { ∞ } \overline {\mathbb {R}} = \mathbb {R} \cup \{\infty \} , obtaining a formulation which allows multiple poles and proving an invariance with respect to R ¯ \overline {\mathbb {R}} -preserving Möbius transformations. We obtain a characterization of Stahl–Totik regularity of a GMP matrix in terms of its matrix elements; as an application, we give a proof of a conjecture of Simon – a Cesàro–Nevai property of regular Jacobi matrices on finite gap sets.more » « less
-
Abstract This paper investigates the spectral properties of Jacobi matrices with limit‐periodic coefficients. We show that generically the spectrum is a Cantor set of zero Lebesgue measure, and the spectral measures are purely singular continuous. For a dense set of limit‐periodic Jacobi matrices, we show that the spectrum is a Cantor set of zero lower box counting dimension while still retaining the singular continuity of the spectral type. We also show how results of this nature can be established by fixing the off‐diagonal coefficients and varying only the diagonal coefficients, and, in a more restricted version, by fixing the diagonal coefficients to be zero and varying only the off‐diagonal coefficients. We apply these results to produce examples of weighted Laplacians on the multidimensional integer lattice having purely singular continuous spectral type and zero‐dimensional spectrum.more » « less
-
Many problems in quantum dynamics can be cast as the decay of a single quantum state into a continuum. The time-dependent overlap with the initial state, called the fidelity, characterizes this decay. We derive an analytic expression for the fidelity after a quench to an ergodic Hamiltonian. The expression is valid for both weak and strong quenches, and timescales before finiteness of the Hilbert space limits the fidelity. It reproduces initial quadratic decay and asymptotic exponential decay with a rate which, for strong quenches, differs from Fermi’s golden rule. The analysis relies on the statistical Jacobi approximation (SJA), which was originally applied in nearly localized systems, and which we here adapt to well-thermalizing systems. Our results demonstrate that the SJA is predictive in disparate regimes of quantum dynamics.more » « less
-
null; null; null; null (Ed.)Many imaging problems can be formulated as inverse problems expressed as finite-dimensional optimization problems. These optimization problems generally consist of minimizing the sum of a data fidelity and regularization terms. In Darbon (SIAM J. Imag. Sci. 8:2268–2293, 2015), Darbon and Meng, (On decomposition models in imaging sciences and multi-time Hamilton-Jacobi partial differential equations, arXiv preprint arXiv:1906.09502, 2019), connections between these optimization problems and (multi-time) Hamilton-Jacobi partial differential equations have been proposed under the convexity assumptions of both the data fidelity and regularization terms. In particular, under these convexity assumptions, some representation formulas for a minimizer can be obtained. From a Bayesian perspective, such a minimizer can be seen as a maximum a posteriori estimator. In this chapter, we consider a certain class of non-convex regularizations and show that similar representation formulas for the minimizer can also be obtained. This is achieved by leveraging min-plus algebra techniques that have been originally developed for solving certain Hamilton-Jacobi partial differential equations arising in optimal control. Note that connections between viscous Hamilton-Jacobi partial differential equations and Bayesian posterior mean estimators with Gaussian data fidelity terms and log-concave priors have been highlighted in Darbon and Langlois, (On Bayesian posterior mean estimators in imaging sciences and Hamilton-Jacobi partial differential equations, arXiv preprint arXiv:2003.05572, 2020). We also present similar results for certain Bayesian posterior mean estimators with Gaussian data fidelity and certain non-log-concave priors using an analogue of min-plus algebra techniques.more » « less
