Eigenvalue excluding for perturbed-periodic one-dimensional Schrödinger operators

Abstract

Subject of investigation in this paper is a one-dimensional Schrödinger equation, where the potential is a sum of a periodic function and a perturbation decaying at ±∞. It is well known that the essential spectrum consists of spectral bands, and that there may or may not be additional eigenvalues below the lowest band or in the gaps between the bands. While enclosures for gap eigenvalues can comparatively easily be obtained from numerical approximations, e.g. by D. Weinstein's bounds, there seems to be no method available so far which is able to exclude eigenvalues in spectral gaps, i.e. which identifies subregions (of a gap) which contain no eigenvalues. Here, we propose such a method. It makes heavy use of computer assistance; nevertheless, the results are completely rigorous in the strict mathematical sense, because all computational errors are taken into account.

Rigorous numerics for NLS: Bound states, spectra, and controllability

“…Although the method relies on computer calculations, the results are mathematically rigorous because all computations are performed in interval arithmetic (using the software package INTLAB [25]), which accounts for all possible rounding errors. In addition to the radii-polynomial technique, there are several other computational methods based on the Contraction Mapping Principle (CMP), such as the Krawczyk operator approach [19,17] or the methods developed by Yamamoto [31], by Koch et al [2], and by Nagatou et al [23]. The main difference is that in the the radii-polynomials approach the enclosure radius r is computed a posteriori and optimally, whereas in most of the other methods an initial guess is made of the set on which T might be contractive, and the hypotheses of the CMP are verified after the fact.…”

“…Λ n := π 2 (2n − 1) 2 + σµ, n > m, respectively. We now introduce the operator T according to (22), (23). (In the de-focusing case σ = −1 the parameter m must be such that π 2 (2m − 1) 2 > σµ to ensure the invertibility of Λ n and, by extension, of A).…”

“…• the exclosure of eigenvalues of the Schrödinger equation with a perturbed periodic potential [22].…”

Fourier–Taylor parameterization of unstable manifolds for parabolic partial differential equations: Formalism, implementation and rigorous validation

“…A thorough review of the literature on computer assisted proof for of PDEs would lead us far afield of the present discussion. We refer to the works of [99,98,94,3,5,7,77,76,66,28,87,10,56] for fuller discussion of computer assisted proof for equilibrium solutions of PDEs, and also [89,64,75,5,24] for more discussion of techniques for validated computation of eigenvalue/eigenvector pairs for infinite dimensional problems. Let us also mention the review articles of [85,73,59] and the book of [83] for broader overview of the field.…”

A functional analytic approach to validated numerics for eigenvalues of delay equations

“…Some works of this kind include the validated numerics for Floquet theory developed in [16], the methods for validated Morse index computations (unstable eigenvalue counts) for infinite dimensional compact maps in [29,37], similar methods for equilibria of parabolic PDEs posed on compact domains in [41,1,44,54,40,55], the validated numerics for stability/instability of traveling waves in [3,2,7,5,6], stability analysis for periodic solutions of delay differential equations [28], the computerassisted proofs of instability for periodic orbits of parabolic partial differential equations found in [22], and the computer-assisted proofs for trapping regions of equilibrium solutions of parabolic PDEs in [18].…”