Asymptotic Dirichlet problem for the Schrödinger operator on negatively curved manifolds

2008 ◽  
Vol 281 (7) ◽  
pp. 971-977
Author(s):  
Seok Woo Kim ◽  
Yong Hah Lee
Keyword(s):  
Dirichlet Problem ◽  
Schrödinger Operator ◽  
Schrodinger Operator ◽  
Curved Manifolds ◽  
Negatively Curved

scholarly journals Dirichlet Problem for the Schrödinger Operator in a Half Space

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Baiyun Su
Keyword(s):  
Dirichlet Problem ◽  
Half Space ◽  
Schrödinger Operator ◽  
Boundary Data ◽  
Poisson Kernel ◽  
Boundary Function ◽  
Poisson Integral ◽  
Schrodinger Operator ◽  
Generalized Poisson ◽  
Continuous Boundary

For continuous boundary data, the modified Poisson integral is used to write solutions to the half space Dirichlet problem for the Schrödinger operator. Meanwhile, a solution of the Poisson integral for any continuous boundary function is also given explicitly by the Poisson integral with the generalized Poisson kernel depending on this boundary function.

Global and Asymptotic Estimates for the Eigenvalues of −Δ‎ + q when q Is Real

2018 ◽  
Author(s):  
D. E. Edmunds ◽  
W. D. Evans
Keyword(s):  
Dirichlet Problem ◽  
Neumann Problem ◽  
Open Subset ◽  
Schrödinger Operator ◽  
General Result ◽  
Asymptotic Estimates ◽  
Schrodinger Operator ◽  
The Neumann Problem

This chapter is devoted to the study of the Schrödinger operator −Δ‎ + q with q real, and, in particular, the distribution of its eigenvalues. A general result is established on an open subset Ω‎ of Rn using the Max–Min Principle and covering families of congruent cubes for the Dirichlet problem and a Whitney covering for the Neumann problem. The Cwikel–Lieb–Rosenbljum inequality is proved for q in Ln/2(Rn).

scholarly journals EIGENFUNCTION EXPANSIONS OF THE MAGNETIC SCHRÖDINGER OPERATOR IN BOUNDED DOMAINS.

2021 ◽  
pp. 5-14
Author(s):  
A.R. Aliev ◽  
◽  
Sh.Sh. Rajabov ◽  
◽  
Keyword(s):  
Dirichlet Problem ◽  
Spectral Parameter ◽  
Schrödinger Operator ◽  
Basis Property ◽  
Multidimensional Case ◽  
Bounded Domains ◽  
Schrodinger Operator ◽  
Magnetic Schrödinger Operator ◽  
Uniqueness Of A Solution ◽  
Generalized Dirichlet

In this work, we introduce the magnetic Schrödinger operator corresponding to the generalized Dirichlet problem. We prove its self-adjointness and discreteness of the spectrum in bounded domains in the multidimensional case. We also prove the basis property of its eigenfunctions in the Lebesgue space and in the magnetic Sobolev space. We give a new characteristic of the definition domain of the magnetic Schrödinger operator. We investigate the existence and uniqueness of a solution of the magnetic Schrödinger equation with a spectral parameter. It is proved that if the spectral parameter is different from the eigenvalues, then the first generalized Dirichlet problem has a unique solution. We then find the solvability condition for the generalized Dirichlet problem when the spectral parameter coincides with the eigenvalue of the Schrödinger magnetic operator.

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