Estimate of fourier transforms with respect to the system of generalized eigenfunctions of the Schrödinger operator with Stummel-type potential

1999 ◽  
Vol 65 (4) ◽  
pp. 454-461
Author(s):  
L. V. Kritskov
Keyword(s):  
Schrödinger Operator ◽  
Fourier Transforms ◽  
Schrodinger Operator ◽  
Generalized Eigenfunctions ◽  
Type Potential

Exponential Localization for Multi-Dimensional Schrödinger Operator with Random Point Potential

1997 ◽  
Vol 09 (04) ◽  
pp. 425-451 ◽  
Author(s):  
Anne Boutet de Monvel ◽  
Vadim Grinshpun
Keyword(s):  
Schrödinger Operator ◽  
Multiscale Analysis ◽  
Infinite Order ◽  
Point Spectrum ◽  
Integrated Density Of States ◽  
Schrodinger Operator ◽  
Negative Part ◽  
Negative Spectrum ◽  
Analysis Scheme ◽  
Generalized Eigenfunctions

We consider a Schrödinger operator -Δα(ω) on L2(ℝd)(d=2,3) whose potential is a sum of point potentials, centered at sites of ℤd, with independent and identically distributed random amplitudes. We prove the existence of the pure point spectrum and the exponential decay of the corresponding eigenfunctions at the negative semi-axis for certain regimes of the disorder. In order to prove localization results, we elucidate the structure of the generalized eigenfunctions of -Δα(ω) and the relation between its negative spectrum and the spectra of a family of infinite-order operators on ℓ2(ℤd). We apply the multiscale analysis scheme to investigate the point spectrum of these operators. We also prove the absolute continuity of the integrated density of states of the operator on the negative part of its spectrum.

scholarly journals Ground State for the Schrödinger Operator with the Weighted Hardy Potential

2011 ◽  
Vol 2011 ◽  
pp. 1-26
Author(s):  
J. Chabrowski ◽  
K. Tintarev
Keyword(s):  
Laplace Operator ◽  
Schrödinger Operator ◽  
Ground State ◽  
Ground States ◽  
Schrodinger Operator ◽  
Hardy Potential ◽  
Higher Integrability ◽  
Hardy Type ◽  
The Laplace Operator ◽  
Type Potential

We establish the existence of ground states on for the Laplace operator involving the Hardy-type potential. This gives rise to the existence of the principal eigenfunctions for the Laplace operator involving weighted Hardy potentials. We also obtain a higher integrability property for the principal eigenfunction. This is used to examine the behaviour of the principal eigenfunction around 0.

scholarly journals On the two-dimensional Schrödinger operator with an attractive potential of the Bessel-Macdonald type

2020 ◽  
pp. 168385
Author(s):  
Wellisson B. De Lima ◽  
Oswaldo M. Del Cima ◽  
Daniel H.T. Franco ◽  
Bruno C. Neves
Keyword(s):  
Schrödinger Operator ◽  
Attractive Potential ◽  
Two Dimensional ◽  
Schrodinger Operator
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