Eigenvalue asymptotics for the Schrödinger operator with perturbed periodic potential

Summary We consider the Schrödinger operatorH=–+W+V acting inL ²( ^m ),m2, with periodic potentialW perturbed by a potentialV which decays slowly at infinity. We study the asymptotic behaviour of the discrete spectrum ofH near any given boundary point of the essential spectrum.Oblatum 1-VII-1991 & 20-I-1992Partly supported by the Bulgarian Science Foundation under contract No MM 8/1991     

Quasilevels of a two-particle Schrödinger operator with a perturbed periodic potential

We consider a two-dimensional periodic Schrödinger operator perturbed by the interaction potential of two one-dimensional particles. We prove that quasilevels (i.e., eigenvalues or resonances) of the given operator exist for a fixed quasimomentum and a small perturbation near the band boundaries of the corresponding periodic operator. We study the asymptotic behavior of the quasilevels as the coupling constant goes to zero. We obtain a simple condition for a quasilevel to be an eigenvalue.     

A remark on the essential self-adjointness of a Schrödinger operator with a singular potential

We examine the operators=–+v, v L_{2, loe} (R ⁿ ), where S satisfies a natural additional condition of a local nature. If a condition of Titchmarsh type is fulfilled at infinity, then S is essentially self-adjoint in L₂(Rⁿ).Translated from Matematicheskie Zametki, Vol. 20, No. 4, pp. 571–580, October, 1976.     

Multidimensional Hardy inequalities and the absence of positive eigenvalues for the Schrödinger operator with a complex potential

The operator –(d/dx)²+ where is a complex function, |J(x)(1+|x|¹⁺|>, has no positive eigenvalues. This follows from the existence of a triangular Green function for the operator –(d/dx)^z–,gl>0, and from Hardy type inequalities. We give the multidimensional analogues of these inequalities and we prove the absence of positive eigenvalues for the operator –+.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 138, pp. 33–34, 1984.     

Absolute continuity of the spectrum of the periodic Schrödinger operator in a layer and in a smooth cylinder

The Schr?dinger operator H = −Δ + V is considered in a layer or in a d-dimensional cylinder. The potential V is assumed to be periodic with respect to a lattice. The absolute continuity of H is established, provided that V ∈ L _p,loc, where p is a real number greater than d/2 in the case of a layer and p > max(d/2, d − 2) for a cylinder. Bibliography: 14 titles.     

On the Riesz-means of negative eigenvalues for a fractional Schrödinger operator

We furnish an upper bound for the moment of negative eigenvalues for a relativistic Schrödinger operator defined in a proper subspace of square-integrable functions.     

Formula for the number of eigenvalues of a three-particle Schrödinger operator on a lattice

We consider a system of three arbitrary quantum particles on a three-dimensional lattice that interact via short-range attractive potentials. We obtain a formula for the number of eigenvalues in an arbitrary interval outside the essential spectrum of the three-particle discrete Schrödinger operator and find a sufficient condition for the discrete spectrum to be finite. We give an example of an application of our results.     

On the density of states for the periodic Schrödinger operator

An asymptotic formula for the density of states of the polyharmonic periodic operator (?δ) ^l +V inR ⁿ ,n≥2,l>1/2 is obtained. Special consideration is given to the case of the Schrödinger equationn=3,l=1,V being a periodic potential, where the second term of the asymptotic is found.     

Positivity of eigenvalues of the two-particle Schrödinger operator on a lattice

We consider the family H(k) of two-particle discrete Schrödinger operators depending on the quasimomentum of a two-particle system k ∈ $\mathbb{T}^d $ , where $\mathbb{T}^d $ is a d-dimensional torus. This family of operators is associated with the Hamiltonian of a system of two arbitrary particles on the d-dimensional lattice ?^d, d ≥ 3, interacting via a short-range attractive pair potential. We prove that the eigenvalues of the Schrödinger operator H(k) below the essential spectrum are positive for all nonzero values of the quasimomentum k ∈ $\mathbb{T}^d $ if the operator H(0) is nonnegative. We establish a similar result for the eigenvalues of the Schrödinger operator H₊(k), k ∈ $\mathbb{T}^d $ , corresponding to a two-particle system with repulsive interaction.     

Complex powers of the Schrödinger operator with a regularly growing potential

A meromorphic extension to the entire plane is obtained for the -function of a Schrödinger operator with a potential that increases at infinity in a power-like manner.Translated from Problemy Matematicheskogo Analiza, No. 11, pp. 176–187, 1990.     

The Schrödinger operator with a weakly accelerating potential

The ideas of scattering theory are applied to the construction of a unitary operator realizing the similarity of the operator - id/d in L₂() with a one-dimensional Schrödinger operator on the semiaxis with potential v(x), admitting at infinity the estimate.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 147, pp. 10–12, 1985.     

Schrödinger operator with a perturbed small steplike potential

We consider the Schrödinger operator with a potential that is periodic with respect to two variables and has the shape of a small step perturbed by a function decreasing with respect to a third variable. We show that under certain conditions on the magnitudes of the step and the perturbation, a unique level that can be an eigenvalue or a resonance exists near the essential spectrum. We find the asymptotic value of this level.     

Absolute continuity of the periodic magnetic Schrödinger operator

We prove that the spectrum of the Schrödinger operator with periodic electric and magnetic potentials is absolutely continuous.     

On infinite-rank singular perturbations of the Schrödinger operator

Schrödinger operators with infinite-rank singular potentials V=Σ _i,j=1 ^∞ b _ij〈φ_j,·〉φ_i are studied under the condition that the singular elements ψ _j are ξ _j(t)-invariant with respect to scaling transformationsin ?³.