AN ABSTRACT RADIATION CONDITION AND APPLICATIONS TO N-BODY SYSTEMS

2000 ◽  
Vol 12 (05) ◽  
pp. 767-803 ◽  
Author(s):  
JACOB SCHACH MØLLER
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Stark Effect ◽  
Schrödinger Operators ◽  
Radiation Condition ◽  
Schrodinger Operators ◽  
Limiting Absorption Principle ◽  
Inhomogeneous Schrödinger Equation

In the setting of Mourre [18] we characterize the "outgoing" and "incoming" solutions, to the abstract inhomogeneous Schrödinger equation (H-E)u=v, given by the Limiting Absorption Principle. The characterization is in terms of an abstract radiation condition and as an application we give a characterization, in the framework of weighted L2-spaces, of the outgoing and incoming solutions for N-body Schrödinger operators with and without Stark effect. The abstract radiation condition translates in the application into a radiation condition considered by Isozaki in [13].

Identities for Eigenvalues of the Schrödinger Equation with Energy-Dependent Potential

2011 ◽  
Vol 66 (12) ◽  
pp. 699-704 ◽  
Author(s):  
Chuan Fu Yang
Keyword(s):  
Schrödinger Equation ◽  
Boundary Conditions ◽  
Schrodinger Equation ◽  
Eigenvalue Problems ◽  
Schrödinger Operators ◽  
Contour Integration ◽  
Schrodinger Operators ◽  
Separated Boundary Conditions ◽  
Dependent Potential ◽  
Energy Dependent

The present paper deals with eigenvalue problems for the Schrödinger equation with energy dependent potential and some separated boundary conditions. Using the method of contour integration, we obtain some new regularized traces for this class of Schrödinger operators.

scholarly journals Remarks on L p -limiting absorption principle of Schrödinger operators and applications to spectral multiplier theorems

2018 ◽  
Vol 30 (1) ◽  
pp. 43-55 ◽  
Author(s):  
Shanlin Huang ◽  
Xiaohua Yao ◽  
Quan Zheng
Keyword(s):  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Spectral Condition ◽  
Multiplier Theorem ◽  
Spectral Multiplier ◽  
Limiting Absorption Principle ◽  
Riesz Operators ◽  
Multiplier Theorems ◽  
Spectral Multiplier Theorems ◽  
Formula Type

Abstract This paper comprises two parts. We first investigate an {L^{p}} -type of limiting absorption principle for Schrödinger operators {H=-\Delta+V} on {\mathbb{R}^{n}} ( {n\geq 3} ), i.e., we prove the ϵ-uniform {L^{{2(n+1)}/({n+3})}} – {L^{{2(n+1)}/({n-1})}} -estimates of the resolvent {(H-\lambda\pm i\epsilon)^{-1}} for all {\lambda>0} under the assumptions that the potential V belongs to some integrable spaces and a spectral condition of H at zero is satisfied. As applications, we establish a sharp Hörmander-type spectral multiplier theorem associated with Schrödinger operators H and deduce {L^{p}} -bounds of the corresponding Bochner–Riesz operators. Next, we consider the fractional Schrödinger operator {H=(-\Delta)^{\alpha}+V} ( {0<2\alpha<n} ) and prove a uniform Hardy–Littlewood–Sobolev inequality for {(-\Delta)^{\alpha}} , which generalizes the corresponding result of Kenig–Ruiz–Sogge [20].

scholarly journals Feynman propagators on static spacetimes

2018 ◽  
Vol 30 (03) ◽  
pp. 1850006 ◽  
Author(s):  
Jan Dereziński ◽  
Daniel Siemssen
Keyword(s):  
Gordon Equation ◽  
Boundary Value ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Electromagnetic Potential ◽  
Limiting Absorption Principle ◽  
Feynman Propagator ◽  
Klein Gordon ◽  
Klein Gordon Equation

We consider the Klein–Gordon equation on a static spacetime and minimally coupled to a static electromagnetic potential. We show that it is essentially self-adjoint on [Formula: see text]. We discuss various distinguished inverses and bisolutions of the Klein–Gordon operator, focusing on the so-called Feynman propagator. We show that the Feynman propagator can be considered the boundary value of the resolvent of the Klein–Gordon operator, in the spirit of the limiting absorption principle known from the theory of Schrödinger operators. We also show that the Feynman propagator is the limit of the inverse of the Wick rotated Klein–Gordon operator.

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