Limiting absorption principle for some Schrödinger operators with exploding potentials, II (original) (raw)

The Absolutely Continuous Spectrum of One-Dimensional Schrödinger Operators with Decaying Potentials

Communications in Mathematical Physics, 1998

We investigate one-dimensional Schrödinger operators with asymptotically small potentials. It will follow from our results that if |V (x)| ≤ C(1+x) −α with α > 1/2, then ac = (0, ∞) is an essential support of the absolutely continuous part of the spectral measure. We also prove that if C := lim sup x→∞ x |V (x)| < ∞, then the spectrum is purely absolutely continuous on ((2C/π) 2 , ∞). These results are optimal.

Résonances réelles et propriétés spectrales de l'opérateur de Schrödinger non-autoadjoint

2020

In this thesis, we study the large-time behavior of solutions to Schrödinger equation with complex-valued potentials. In the first part, we are interested in rapidly decreasing potentials. We establish the resolvent expansions at threshold and near positive resonances. We obtain the expansions in time of solutions under different conditions, including the existence of positive resonances and zero resonance or/and zero eigenvalue. In the second part, we are interested in slowly decreasing potentials. We establish Gevrey estimates for the resolvent and the large-time expansions for Schrödinger and heat semi-groups with sub-exponential time-decay estimates on the remainder. These results generalize the results of X. P. Wang to potentials satisfying a virial condition at infinity. Our results in the two parts cover the case of zero eigenvalue of arbitrary geometric multiplicity.Dans cette thèse, on étudie le comportement en temps grand des solutions de l’équation de Schrödinger avec pot...

Essential spectra of pseudodifferential operators and exponential decay of their solutions. Applications to Schrödinger operators

2008

The aim of this paper is to study relations between the location of the esse:1tial spectrum and the exponential decay of eigenfunctions of pseudodifferential operators on LP (]Rn) perturbed by singular potentials. Our approach to this problem is via the limit operators method. This method associates with each band-dominated operator A a family op(A) of so-callee. limit operators which reflect the properties of A at infinity. Consider the compactification of]Rn by the "infinitely distant" sphere Sn-l. Then the set op(A) can be written as the union of its components 0P'r/w (A) where w runs through the points of sn-l and where 0P'r/w (A) collects all limit operators of A whidu reflect the properties of A if one tends to infinity "in the direction of w". Set sP'r/wA := UAhEOP7)w(A)SpAh. W,? show that the distance of an eigenvalue A t/:. sPessA to SP'r/w A determines the exponential decay of the A-eigenfunctions of A in the direction of w. W,? apply these results to estimate the exponential decay of eigenfunctions of electromagnetic Schrodinger operators for a large class of electric potenticJs, in particular, for multiparticle Schrodinger operators and periodic Schrodillger operators perturbed by slowly oscillating at infinity potentials.

On the semi-classical analysis of Schrödinger operators with linear electric potentials on a bounded domain

Asymptotic Analysis

The aim of this paper is to establish the asymptotic expansion of the eigenvalues of the Stark Hamiltonian, with a strong uniform electric field and Dirichlet boundary conditions on a smooth bounded domain of R N , N ⩾ 2. This work aims at generalizing the recent results of Cornean, Krejčiřik, Pedersen, Raymond, and Stockmeyer in dimension 2. More precisely, in dimension N, in the strong electric field limit, we derive, under certain local convexity conditions, a full asymptotic expansion of the low-lying eigenvalues. To establish our main result, we perform the construction of quasi-modes. The “optimality” of our constructions is then established thanks to a reduction to model operators and localization estimates.

On Spectral Theory for Schrödinger Operators with Operator-Valued Potentials

2013

Given a complex, separable Hilbert space , we consider differential expressions of the type τ = - (d^2/dx^2) + V(x), with x ∈ (a,∞) or x ∈. Here V denotes a bounded operator-valued potential V(·) ∈() such that V(·) is weakly measurable and the operator norm V(·)_() is locally integrable. We consider self-adjoint half-line L^2-realizations H_α in L^2((a,∞); dx; ) associated with τ, assuming a to be a regular endpoint necessitating a boundary condition of the type sin(α)u'(a) + cos(α)u(a)=0, indexed by the self-adjoint operator α = α^* ∈(). In addition, we study self-adjoint full-line L^2-realizations H of τ in L^2(; dx; ). In either case we treat in detail basic spectral theory associated with H_α and H, including Weyl--Titchmarsh theory, Green's function structure, eigenfunction expansions, diagonalization, and a version of the spectral theorem.

Limiting absorption principle for the Dirac operator

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