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     

Essential self-adjointness of the Schrödinger operator in a magnetic field

We consider the multidimensional Schrödinger operator in an electromagnetic field. Under certain Stummel-type conditions imposed on the magnetic and electric potentials, we prove the essential self-adjointness of the magnetic Schrödinger operator.     

Quantum scattering near the lowest Landau threshold for a Schrödinger operator with a constant magnetic field

For fixed magnetic quantum number m results on spectral properties and scattering theory are given for the three-dimensional Schrödinger operator with a constant magnetic field and an axisymmetrical electric potential V. In various, mostly singular settings, asymptotic expansions for the resolvent of the Hamiltonian H _m+H_om+V are deduced as the spectral parameter tends to the lowest Landau threshold. Furthermore, scattering theory for the pair (H _m, H _om) is established and asymptotic expansions of the scattering matrix are derived as the energy parameter tends to the lowest Landau threshold.     

Generalized Liouville property for Schrödinger operator on Riemannian manifolds

In this paper, we prove that the dimension of the space of positive (bounded, respectively) -harmonic functions on a complete Riemannian manifold with -regular ends is equal to the number of ends (-nonparabolic ends, respectively). This result is a solution of an open problem of Grigor'yan related to the Liouville property for the Schr?dinger operator . We also prove that if a given complete Riemannian manifold is roughly isometric to a complete Riemannian manifold satisfying the volume doubling condition, the Poincaré inequality and the finite covering condition on each end, then the dimension of the space of positive (bounded, respectively,) solutions for the Schr?dinger operator with a potential satisfying a certain decay rate on the manifold is equal to the number of ends (-nonparabolic ends, respectively). This is a partial answer of the question, suggested by Li, related to the regularity of ends of a complete Riemannian manifold. Especially, our results directly generalize various earlier results of Yau, of Li and Tam, of Grigor'yan, and of present authors, but with different techniques that the peculiarity of the Schr?dinger operator demands. Received: 4 April 2000; in final form: 19 September 2000 / Published online: 25 June 2001     

Semi-classical resolvent estimates for the Schrödinger operator on non-compact complete Riemannian manifolds

We prove uniform semi-classical estimates for the resolvent of the Schrödinger operator h ² _g + V (x), 0 < h 1, at a nontrapping energy level E > 0, where V is a real-valued non-negative potential and _g denotes the positive Laplace-Beltrami operator on a non-compact complete Riemannian manifold which may have a nonempty compact smooth boundary.*Partially supported by CNPq (Brazil)     

Spectrum and the trace formula for a two-dimensional Schrödinger operator in a homogeneous magnetic field

In the space L ₂(?²), we consider the operator

$H = \left( {\frac{1}{i}\frac{\partial }{{\partial x_1 }} - x_2 } \right)^2 + \left( {\frac{1}{i}\frac{\partial }{{\partial x_2 }} + x_1 } \right)^2 + V,V = V(x) \in L_2 (\mathbb{R}^2 ).$

. We study the spectrum of H and, for V ∈ C ₀ ² (?²), prove the trace formula

$\sum\limits_{k = 0}^\infty {\left( {\sum\limits_{i = - k}^\infty {(4k + 2 - \mu _k^{(i)} ) + c_0 } } \right)} = \frac{1}{{8\pi }}\int\limits_{\mathbb{R}^2 } {V^2 (x)dx,} $

where c ₀ = π ^?1 \(\smallint _{\mathbb{R}^2 } \) V(x) dx and the µ _k ⁽ⁱ⁾ are the eigenvalues of H.

     

Soliton dynamics for the nonlinear Schrödinger equation with magnetic field

The semiclassical regime of a nonlinear focusing Schrödinger equation in presence of non-constant electric and magnetic potentials V, A is studied by taking as initial datum the ground state solution of an associated autonomous stationary equation. The concentration curve of the solutions is a parameterization of the solutions of the second order ordinary equation \({\ddot x=-\nabla V(x)-\dot x\times B(x)}\), where \({B=\nabla\times A}\) is the magnetic field of a given magnetic potential A.     

Time analyticity for the parabolic type Schrödinger equation on Riemannian manifold with integral Ricci curvature condition

In the paper, we investigate the pointwise time analyticity of the parabolic type Schrödinger equation on a complete Riemannian manifold with integral Ricci curvature condition.     

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.     

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).     

A discrete Schrödinger operator on a graph

We consider the discrete Schrödinger operator on the graph obtained in the strong-coupling approximation from the standard electron Schrödinger operator in the system composed of a quantum wire and quantum dot. We investigate the general spectral properties of this operator and the problem of the existence and behavior of the eigenvalues and resonances depending on the small coupling constant. We study the scattering problem for weak potentials in the stationary approach.     

Stability Estimates for the Lowest Eigenvalue of a Schrödinger Operator

There is a family of potentials that minimize the lowest eigenvalue of a Schrödinger operator under the constraint of a given L ^p norm of the potential. We give effective estimates for the amount by which the eigenvalue increases when the potential is not one of these optimal potentials. Our results are analogous to those for the isoperimetric problem and the Sobolev inequality. We also prove a stability estimate for Hölder’s inequality, which we believe to be new.     

Trace formulas for a discrete Schrödinger operator

The Schrödinger operator with complex decaying potential on a lattice is considered. Trace formulas are derived on the basis of classical results of complex analysis. These formulas are applied to obtain global estimates of all zeros of the Fredholm determinant in terms of the potential.     

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.     

Essential selfadjointness of singular magnetic Schrödinger operators on Riemannian manifolds

In this paper we extend the well-known Leinfelder–Simader theorem on the essential selfadjointness of singular Schrödinger operators to arbitrary complete Riemannian manifolds. This improves some earlier results of Shubin, Milatovic and others.