On solvability of the Dirichlet problem to the semilinear Schrödinger equation with singular potential

2007 ◽  
Vol 143 (2) ◽  
pp. 2857-2868 ◽  
Author(s):  
A. V. Demyanov ◽  
A. I. Nazarov
Keyword(s):  
Dirichlet Problem ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Singular Potential ◽  
Semilinear Schrödinger Equation

On the semilinear Schrödinger equation with time dependent coefficients

2014 ◽  
Vol 287 (17-18) ◽  
pp. 1986-2001
Author(s):  
Takuya Gonda ◽  
Shuji Machihara ◽  
Tohru Ozawa
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Time Dependent ◽  
Semilinear Schrödinger Equation

scholarly journals Evolution by Schrödinger equation of Aharonov–Berry superoscillations in centrifugal potential

2019 ◽  
Vol 475 (2225) ◽  
pp. 20180390 ◽  
Author(s):  
F. Colombo ◽  
J. Gantner ◽  
D. C. Struppa
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Singular Potential ◽  
Singular Potentials ◽  
Limit Solution ◽  
Quadratic Hamiltonian

In recent years, we have investigated the evolution of superoscillations under Schrödinger equation with non-singular potentials. In all those cases, we have shown that superoscillations persist in time. In this paper, we investigate the centrifugal potential, which is a singular potential, and we show that the techniques developed to study the evolution of superoscillations in the case of the Schrödinger equation with a quadratic Hamiltonian apply to this setting. We also specify, in the case of the centrifugal potential, the notion of super-shift of the limit solution, a fact explained in the last section of this paper. It then becomes apparent that superoscillations are just a particular case of super-shift.

scholarly journals ON THE LOCATION OF SPIKES FOR THE SCHRÖDINGER EQUATION WITH ELECTROMAGNETIC FIELD

2005 ◽  
Vol 07 (02) ◽  
pp. 251-268 ◽  
Author(s):  
SIMONE SECCHI ◽  
MARCO SQUASSINA
Keyword(s):  
Electromagnetic Field ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Necessary Conditions ◽  
Three Dimensional ◽  
External Electromagnetic Field ◽  
Power Type ◽  
Wave Solutions ◽  
Semilinear Schrödinger Equation ◽  
Ground Energy

We consider the standing wave solutions of the three dimensional semilinear Schrödinger equation with competing potential functions V and K and under the action of an external electromagnetic field B. We establish some necessary conditions for a sequence of such solutions to concentrate, in two different senses, around a given point. In the particular but important case of nonlinearities of power type, the spikes locate at the critical points of a smooth ground energy map independent of B.

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