scholarly journals On local‐in‐time Strichartz estimates for the Schrödinger equation with singular potentials

2021 ◽  
Author(s):  
Seongyeon Kim ◽  
Ihyeok Seo ◽  
Jihyeon Seok
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Strichartz Estimates ◽  
Singular Potentials

scholarly journals COUNTEREXAMPLES TO STRICHARTZ ESTIMATES FOR THE MAGNETIC SCHRÖDINGER EQUATION

2011 ◽  
Vol 13 (02) ◽  
pp. 213-234 ◽  
Author(s):  
LUCA FANELLI ◽  
ANDONI GARCIA
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Space Dimension ◽  
Strichartz Estimates

In space dimension n ≥ 3, we consider the magnetic Schrödinger Hamiltonian H = -(∇ - iA(x))2and the corresponding Schrödinger equation [Formula: see text] We show some explicit examples of potentials A, with less than Coulomb decay, for which any solution of this equation cannot satisfy Strichartz estimates, in the whole range of Schrödinger admissibility.

SPHERICAL AVERAGED ENDPOINT STRICHARTZ ESTIMATES FOR THE TWO-DIMENSIONAL SCHRÖDINGER EQUATION WITH INVERSE SQUARE POTENTIAL

2010 ◽  
Vol 07 (03) ◽  
pp. 365-382 ◽  
Author(s):  
I-KUN CHEN
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Strichartz Estimate ◽  
Strichartz Estimates ◽  
Angular Variable ◽  
Two Dimensional ◽  
Radial Variable

We investigate the two-dimensional Schrödinger equation with repulsive inverse square potential, and we prove the following homogeneous endpoint Strichartz estimate: [Formula: see text] where [Formula: see text] is a norm that applies L2average on the angular variable, first, and then the supremum on the radial variable.

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.

Sign in / Sign up

Export Citation Format

Share Document