Classes of uniform convergence of spectral expansions for the one-dimensional Schrödinger operator with a distribution potential

For the self-adjoint Schrödinger operator ? defined on ? by the differential operation ?(d/dx)² + q(x) with a distribution potential q(x) uniformly locally belonging to the space W ₂ ^?1, we describe classes of functions whose spectral expansions corresponding to the operator ? absolutely and uniformly converge on the entire line ?. We characterize the sharp convergence rate of the spectral expansion of a function using a two-sided estimate obtained in the paper for its generalized Fourier transforms.     

Negative powers of a singular Schrödinger operator and convergence of spectral decompositions

We compare theL ₂( ^N )-norms of negative powers of various Laplace and Schrödinger operators possessing a singular potential whose singularities lie on some manifolds. We write out sufficient conditions for uniform convergence and localization of spectral decompositions of functions from the Liouville class.Translated fromMatematicheskie Zametki, Vol. 59, No. 3, pp. 428–436, March, 1996.The author wishes to express deep gratitude to Prof. Sh. A. Alimov for his attention to this work.     

An inverse spectral problem for a fractional Schrödinger operator

Archiv der Mathematik - We establish that the potential appearing in a fractional Schrödinger operator is uniquely determined by an internal spectral data.     

Semiclassical spectral series of the Schrödinger operator with a delta potential on a straight line and on a sphere

We describe the spectral series of the Schrödinger operator H = ?(h²/2)Δ + V(x) + αδ(x?x₀), α ∈ ?, with a delta potential on the real line and on the three- and two-dimensional standard spheres in the semiclassical limit as h → 0. We consider a smooth potential V(x) such that lim_|x|→∞V(x)=+∞ in the first case and V(x) = 0 in the last two cases. In the semiclassical limit in each case, we describe the classical trajectories corresponding to the quantum problem with a delta potential.     

The resolvent equation of the one-dimensional Schrödinger operator on the whole axis

Under certain conditions on the magnetic and electric potentials, we prove that the corresponding one-dimensional magnetic Schrödinger operator on the whole axis is selfadjoint and establish that Fredholm theory is applicable to the resolvent equation of this operator.     

Spectrum of the two-particle Schrödinger operator on a lattice

We consider the family of two-particle discrete Schrödinger operators H(k) associated with the Hamiltonian of a system of two fermions on a ν-dimensional lattice ?, ν ≥, 1, where k ∈ \(\mathbb{T}^\nu \) ≡ (? π, π]^ν is a two-particle quasimomentum. We prove that the operator H(k), k ∈ \(\mathbb{T}^\nu \), k ≠ 0, has an eigenvalue to the left of the essential spectrum for any dimension ν = 1, 2, ... if the operator H(0) has a virtual level (ν = 1, 2) or an eigenvalue (ν ≥ 3) at the bottom of the essential spectrum (of the two-particle continuum).     

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.     

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.     

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.     

The convergence of fourier series in eigenfunctions of the Schrödinger operator with Kato potential

We obtain sharp conditions for the absolute uniform convergence of Fourier series in the eigenfunctions of the Schrödinger operator with Kato potential in a bounded domain for functions lying in the domains of generalized fractional powers of the original Schrödinger operator or in generalized Besov classes with a sharp exponent.     

Asymptotic expansion of the integrated density of states of a two-dimensional periodic Schrödinger operator

We prove the complete asymptotic expansion of the integrated density of states of a two-dimensional Schrödinger operator with a smooth periodic potential.     

Eigenvalue problems for the Schrödinger operator with the magnetic field on a compact Riemannian manifold

We consider the eigenvalue problem of the Schrödinger operator with the magnetic field on a compact Riemannian manifold. First we discuss the least eigenvalue. We give a representation of the least eigenvalue by the variational formula and give a relation to the least eigenvalue of the Schrödinger operator without the magnetic field. Second, we discuss the asymptotic distribution of eigenvalues by obtaining the asymptotic expansion of the kernel of semigroup. Here we use the theory of asymptotic expansion for Wiener functionals.     

Lower bounds on the eigenvalue sums of the Schrödinger operator and the spectral conservation law

We consider the Schr?dinger operator H = −Δ − V(x), V > 0, acting in the space L² (\mathbbR^d) L^2 (\mathbb{R}^d) and study relations between the behavior of V at infinity and properties of the negative spectrum of H. Bibliography: 34 titles.     

On the essential spectrum of a four-particle Schrödinger operator on a lattice

The Hamiltonian of a system of four arbitrary quantum particles with three-particle short-range interaction potentials on a three-dimensional lattice is examined. The location of the essential spectrum of this Hamiltonian is described by Faddeev’s equations.     

On an isoperimetric inequality for a Schrödinger operator depending on the curvature of a loop

Let y be a smooth closed curve of length 2π in ?³, and let κ(s) be its curvature, regarded as a function of arc length. We associate with this curve the one-dimensional Schrödinger operator $H_\gamma = - \tfrac{{d^2 }}{{ds^2 }} + \kappa ^2 (s)$ acting on the space of square integrable 2π-periodic functions. A natural conjecture is that the lowest spectral value e₀ (y) of H_y is bounded below by 1 for any y (this value is assumed when y is a circle). We study a family of curves y that includes the circle and for which e₀(y) = 1 as well. We show that the curves in this family are local minimizers, i.e., e₀(y) can only increase under small perturbations leading away from the family. To our knowledge, the full conjecture remains open.     

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.