scholarly journals Small-energy analysis for the self-adjoint matrix Schrödinger operator on the half line

2011 ◽  
Vol 52 (10) ◽  
pp. 102101 ◽  
Author(s):  
Tuncay Aktosun ◽  
Martin Klaus ◽  
Ricardo Weder
Keyword(s):  
Schrödinger Operator ◽  
Energy Analysis ◽  
The Self ◽  
Half Line ◽  
Schrodinger Operator ◽  
Small Energy ◽  
Adjoint Matrix

scholarly journals A theorem on “localized” self-adjointness of Shrödinger operators withLLOC1-potentials

1982 ◽  
Vol 5 (3) ◽  
pp. 545-552 ◽  
Author(s):  
Hans L. Cycon
Keyword(s):  
Schrödinger Operator ◽  
Schrödinger Operators ◽  
The Self ◽  
Schrodinger Operators ◽  
Schrodinger Operator

We prove a result which concludes the self-adjointness of a Schrödinger operator from the self-adjointness of the associated “localized” Schrödinger operators havingLLOC1-Potentials.

RCD*(K,N) Spaces and the Geometry of Multi-Particle Schrödinger Semigroups

2020 ◽  
Author(s):  
Batu Güneysu
Keyword(s):  
Perturbation Theory ◽  
Schrödinger Operator ◽  
The Self ◽  
Schrodinger Operator ◽  
Hölder Continuous ◽  
Kato Class ◽  
Coupling Methods

Abstract Dedicated to the memory of Kazumasa Kuwada. Let $(X,\mathfrak{d},{\mathfrak{m}})$ be an $\textrm{RCD}^*(K,N)$ space for some $K\in{\mathbb{R}}$, $N\in [1,\infty )$, and let $H$ be the self-adjoint Laplacian induced by the underlying Cheeger form. Given $\alpha \in [0,1]$, we introduce the $\alpha$-Kato class of potentials on $(X,\mathfrak{d},{\mathfrak{m}})$, and given a potential $V:X\to{\mathbb{R}}$ in this class, we denote with $H_V$ the natural self-adjoint realization of the Schrödinger operator $H+V$ in $L^2(X,{\mathfrak{m}})$. We use Brownian coupling methods and perturbation theory to prove that for all $t>0$, there exists an explicitly given constant $A(V,K,\alpha ,t)<\infty$, such that for all $\Psi \in L^{\infty }(X,{\mathfrak{m}})$, $x,y\in X$ one has $$\begin{align*}\big|e^{-tH_V}\Psi(x)-e^{-tH_V}\Psi(y)\big|\leq A(V,K,\alpha,t) \|\Psi\|_{L^{\infty}}\mathfrak{d}(x,y)^{\alpha}.\end{align*}$$In particular, all $L^{\infty }$-eigenfunctions of $H_V$ are globally $\alpha$-Hölder continuous. This result applies to multi-particle Schrödinger semigroups and, by the explicitness of the Hölder constants, sheds some light into the geometry of such operators.

EXISTENCE OF ALMOST EXPONENTIALLY DECAYING STATES FOR BARRIER POTENTIALS

2001 ◽  
Vol 13 (03) ◽  
pp. 267-305 ◽  
Author(s):  
RICHARD LAVINE
Keyword(s):  
Error Estimate ◽  
Half Plane ◽  
Imaginary Part ◽  
Schrödinger Operator ◽  
Small Error ◽  
Lower Half ◽  
Half Line ◽  
Schrodinger Operator ◽  
Lower Half Plane ◽  
Decaying States

For a Schrödinger operator H on the half line whose potential has a trapping barrier, and is convex outside the barrier, there exists a φ, supported mostly inside the barrier, such that for t>0, <φ, e-iHtφ>~e-izt up to a small error, where φ is obtained by cutting off a nonnormalizable solution ψ of Hψ=zψ, and z is in the lower half-plane. The imaginary part of z is estimated explicitly, and the error estimate is explicitly proportional to | Im z log | Im z‖.

Sign in / Sign up

Export Citation Format

Share Document