Absolutely continuous spectrum for one-dimensional Schrödinger operators with slowly decaying potentials: Some optimal results

The absolutely continuous spectrum of one-dimensional Schrödinger operators is proved to be stable under perturbation by potentials satisfying mild decay conditions. In particular, the absolutely continuous spectra of free and periodic Schrödinger operators are preserved under all perturbations satisfying This result is optimal in the power scale. Slightly more general classes of perturbing potentials are also treated. A general criterion for stability of the absolutely continuous spectrum of one-dimensional Schrödinger operators is established. In all cases analyzed, the main term of the asymptotic behavior of the generalized eigenfunctions is shown to have WKB form for almost all energies. The proofs rely on maximal function and norm estimates, and on almost everywhere convergence results for certain multilinear integral operators.

     

Absolute Continuity of a Two-Dimensional Magnetic Periodic Schrödinger Operator ith Potentials of the Type of Measure Derivative

A two-dimensional magnetic periodic Schrödinger operator with a variable metric is considered. An electric potential is assumed to be a distribution formally given by an expression , where d is a periodic signed measure with a locally finite variation. We also assume that the perturbation generated by the electric potential is strongly subject (in the sense of forms) to the free operator. Under this natural assumption, we prove that the spectrum of the Schrödinger operator is absolutely continuous. Bibliography: 15 titles.     

Oscillation of the perturbed Hill equation and the lower spectrum of radially periodic Schrödinger operators in the plane

Generalizing the classical result of Kneser, we show that the
Sturm-Liouville equation with periodic coefficients and an added perturbation term is oscillatory or non-oscillatory (for ) at the infimum of the essential spectrum, depending on whether surpasses or stays below a critical threshold. An explicit characterization of this threshold value is given. Then this oscillation criterion is applied to the spectral analysis of two-dimensional rotation symmetric Schrödinger operators with radially periodic potentials, revealing the surprising fact that (except in the trivial case of a constant potential) these operators always have infinitely many eigenvalues below the essential spectrum.

     

Quasiexact Solution of a Relativistic Finite-Difference Analogue of the Schrödinger Equation for a Rectangular Potential Well

We consider a well-posed formulation of the spectral problem for a relativistic analogue of the one-dimensional Schrödinger equation with differential operators replaced with operators of finite purely imaginary argument shifts exp(±id/dx). We find effective solution methods that permit determining the spectrum and investigating the properties of wave functions in a wide parameter range for this problem in the case of potentials of the type of a rectangular well. We show that the properties of solutions of these equations depend essentially on the relation between and the parameters of the potential and a situation in which the solution for 1 is nevertheless fundamentally different from its Schrödinger analogue is quite possible.     

Bound states of discrete Schrödinger operators with super-critical inverse square potentials

We consider discrete one-dimensional Schrödinger operators whose potentials decay asymptotically like an inverse square. In the super-critical case, where there are infinitely many discrete eigenvalues, we compute precise asymptotics of the number of eigenvalues below a given energy as this energy tends to the bottom of the essential spectrum.

     

The complex germ method in fock space. I. Wave packet type asymptotics

In this paper, we establish a new method of constructing approximate solutions to secondary-quantized equations, for instance, for many-particle Schrödinger and Liouville equations written in terms of the creation and annihilation operators, and also for equations of quantum field theory. The method is based on transformation of these equations to an infinite-dimensional Schrödinger equation, which is investigated by semiclassical methods. We use, and generalize to the infinite-dimensional case, the complex germ method, which yields wave packet type asymptotics in the Schrödinger representation. We find the corresponding asymptotics in the Fock space and show that the state vectors obtained are actually asymptotic solutions to secondary-quantized equations with an accuracy of O(^M/2), M N, with respect to the parameter of the semiclassical expansion.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 2, pp. 310–329, August, 1995.     

Singular Schrödinger Operators as Limits Point Interaction Hamiltonians

In this paper we give results on the approximation of (generalized) Schrödinger operators of the form - + µ for some finite Radon measure µ on R^d. For d = 1 we shall show that weak convergence of measures µ_n to µ implies norm resolvent convergence of the operators - + µ_n to - + µ. In particular Schrödinger operators of the form - + µ for some finite Radon measure µ can be regularized or approximated by Hamiltonians describing point interactions. For d = 3 we shall show that a fairly large class of singular interactions can be regarded as limit of point interactions.     

One Version of the Bethe–Sommerfeld Conjecture

We consider the two-dimensional periodic Schrödinger operator under the assumption that the electric potential contains a term proportional to the -function concentrated on a periodic system of orthogonal lines. For this operator we confirm the Bethe–Sommerfeld conjecture and study the asymptotic behavior of the integrated density of states. We prove that the -potential can be chosen in such a way that the spectrum of the operator contains any given number of gaps. Bibliography: 9 titles.     

The Cauchy problem for the fourth-order nonlinear Schrödinger equation related to the vortex filament

The Cauchy problem of one-dimensional fourth-order nonlinear Schrödinger equation related to the vortex filament is studied. Local well-posedness for initial data in is obtained by the Fourier restriction norm method under certain coefficient condition.     

Semiclassical Asymptotics and Gaps in the Spectra of Magnetic Schrödinger Operators

In this paper, we study an L ² version of the semiclassical approximation of magnetic Schrödinger operators with invariant Morse type potentials on covering spaces of compact manifolds. In particular, we are able to establish the existence of an arbitrary large number of gaps in the spectrum of these operators, in the semiclassical limit as the coupling constant goes to zero.     

An efficient algorithm for the Schrödinger–Poisson eigenvalue problem

We present a new implementation of the two-grid method for computing extremum eigenpairs of self-adjoint partial differential operators with periodic boundary conditions. A novel two-grid centered difference method is proposed for the numerical solutions of the nonlinear Schrödinger–Poisson (SP) eigenvalue problem.We solve the Poisson equation to obtain the nonlinear potential for the nonlinear Schrödinger eigenvalue problem, and use the block Lanczos method to compute the first k   eigenpairs of the Schrödinger eigenvalue problem until they converge on the coarse grid. Then we perform a few conjugate gradient iterations to solve each symmetric positive definite linear system for the approximate eigenvector on the fine grid. The Rayleigh quotient iteration is exploited to improve the accuracy of the eigenpairs on the fine grid. Our numerical results show how the first few eigenpairs of the Schrödinger eigenvalue problem are affected by the dopant in the Schrödinger–Poisson (SP) system. Moreover, the convergence rate of eigenvalue computations on the fine grid is O(h³)$\mathrm{O}(h^3)$.     

Sur la multiplicité de la première valeur propre de l'opérateur de Schrödinger avec champ magnétique sur la sphère

L'objet de cet article est d'étudier la multiplicité de la première valeur propre de l'opérateur de Schrödinger avec champ magnétique sur la sphère , et, répondant en cela à une question posée par Y. Colin de Verdière, de montrer d'une part que cette multiplicité peut être arbitrairement grande, mais que, d'autre part, elle est toujours bornée en fonction de la courbure de la connexion associée.
ABSTRACT. The purpose of this text is to study the first eigenvalue of the Schrödinger operator with magnetic field on the 2-sphere and to show that its multiplicity can be arbitrarily high. We also show that this multiplicity is bounded in terms of the curvature of the corresponding connection. This answers a question asked by Y. Colin de Verdière.

     

Note on ground states of a nonlinear Schrödinger system

We give a sufficient condition for the existence of positive radial ground states of the time-independent Schrödinger system

     

On the location of the essential spectrum of Schrödinger operators

We give estimates on the bottom of the essential spectrum of Schrödinger operators in .

     

Scattering and Wave Operators for One-Dimensional Schrödinger Operators with Slowly Decaying Nonsmooth Potentials

We prove existence of modified wave operators for one-dimensional Schrödinger equations with potential in If in addition the potential is conditionally integrable, then the usual Möller wave operators exist. We also prove asymptotic completeness of these wave operators for some classes of random potentials, and for almost every boundary condition for any given potential.     

Asymptotics of eigenvalue clusters for Schrödinger operators on the Sierpinski gasket

In this note we investigate the asymptotic behavior of spectra of Schrödinger operators with continuous potential on the Sierpinski gasket . In particular, using the existence of localized eigenfunctions for the Laplacian on we show that the eigenvalues of the Schrödinger operator break into clusters around certain eigenvalues of the Laplacian. Moreover, we prove that the characteristic measure of these clusters converges to a measure. Results similar to ours were first observed by A. Weinstein and V. Guillemin for Schrödinger operators on compact Riemannian manifolds.

     

The approximate spectral projection method

In this paper we develop a general method for investigating the spectral asymptotics for various differential and pseudo-differential operators and their boundary value problems, and consider many of the problems posed when this method is applied to mathematical physics and mechanics. Among these problems are the Schrödinger operator with growing, decreasing and degenerating potential, the Dirac operator with decreasing potential, the quasi-classical spectral asymptotics for Schrödinger and Dirac operators, the linearized Navier-Stokes equation, the Maxwell system, the system of reactor kinetics, the eigenfrequency problems of shell theory, and so on. The method allows us to compute the principal term of the spectral asymptotics (and, in the case of Douglis-Nirenberg elliptic operators, also their following terms) with the remainder estimate close to that for the sharp remainder.     

Kato-type estimates for NLS equation in a scalar field and unique solvability of NKGS system in energy space

We consider Schrödinger equation in R²⁺¹${\mathbb{R}}^{2+1}$ with nonlinear scalar potential. The potentials are time-independent or determined as solutions to inhomogeneous wave equations. By constructing a modified propagator, we derive Kato-type smoothing estimates for the nonlinear Schrödinger (NLS) equation. With the help of these results, we prove the unique solvability of the nonlinear Klein–Gordon–Schrödinger (NKGS) system for all time in the energy space.     

On the essential spectrum of Schrödinger operators on Riemannian manifolds

The main result of this paper is a lower bound for the essential spectrum of Schrödinger operators −Δ+V on Riemannian manifolds. In particular, we obtain conditions on V which imply the discreteness of the spectrum, or equivalently, the compactness of the resolvent.     

Imbedded singular continuous spectrum for Schrödinger operators

We construct examples of potentials satisfying where the function is growing arbitrarily slowly, such that the corresponding Schrödinger operator has an imbedded singular continuous spectrum. This solves one of the fifteen ``twenty-first century" problems for Schrödinger operators posed by Barry Simon. The construction also provides the first example of a Schrödinger operator for which Möller wave operators exist but are not asymptotically complete due to the presence of a singular continuous spectrum. We also prove that if the singular continuous spectrum is empty. Therefore our result is sharp.