Ground States for a Pseudo-Relativistic Schrödinger Equation

2021 ◽  
pp. 107-144
Author(s):  
Vincenzo Ambrosio
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Ground States

scholarly journals Non-Kirchhoff Vertices and Nonlinear Schrödinger Ground States on Graphs

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 617
Author(s):  
Riccardo Adami ◽  
Filippo Boni ◽  
Alice Ruighi
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Ground States ◽  
Matching Condition ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger

We review some recent results and announce some new ones on the problem of the existence of ground states for the Nonlinear Schrödinger Equation on graphs endowed with vertices where the matching condition, instead of being free (or Kirchhoff’s), is non-trivially interacting. This category includes Dirac’s delta conditions, delta prime, Fülöp-Tsutsui, and others.

Ground states of a nonlinear drifting Schrödinger equation

2020 ◽  
Vol 105 ◽  
pp. 106324 ◽  
Author(s):  
Li Ma ◽  
Kaiqiang Zhang
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Ground States

scholarly journals Computing the Ground and First Excited States of the Fractional Schrödinger Equation in an Infinite Potential Well

2015 ◽  
Vol 18 (2) ◽  
pp. 321-350 ◽  
Author(s):  
Siwei Duo ◽  
Yanzhi Zhang
Keyword(s):  
Schrödinger Equation ◽  
Excited States ◽  
Schrodinger Equation ◽  
Fractional Laplacian ◽  
Ground States ◽  
Euler Method ◽  
Potential Well ◽  
Fractional Schrödinger Equation ◽  
Fractional Schrodinger Equation ◽  
Infinite Potential Well

AbstractIn this paper, we numerically study the ground and first excited states of the fractional Schrödinger equation in an infinite potential well. Due to the nonlocality of the fractional Laplacian, it is challenging to find the eigenvalues and eigenfunctions of the fractional Schrödinger equation analytically. We first introduce a normalized fractional gradient flow and then discretize it by a quadrature rule method in space and the semi-implicit Euler method in time. Our numerical results suggest that the eigenfunctions of the fractional Schrödinger equation in an infinite potential well differ from those of the standard (non-fractional) Schrödinger equation. We find that the strong nonlocal interactions represented by the fractional Laplacian can lead to a large scattering of particles inside of the potential well. Compared to the ground states, the scattering of particles in the first excited states is larger. Furthermore, boundary layers emerge in the ground states and additionally inner layers exist in the first excited states of the fractional nonlinear Schrödinger equation. Our simulated eigenvalues are consistent with the lower and upper bound estimates in the literature.

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