Logarithmic Schrödinger Equations in Infinite Dimensions

Abstract

We study the logarithmic Schrödinger equation with finite range potential on ${\mathbb{R}}^{{\mathbb{Z}}^d}$. Through a ground-state representation, we associate and construct a global Gibbs measure and show that it satisfies a logarithmic Sobolev inequality. We find estimates on the solutions in arbitrary dimension and prove the existence of weak solutions to the infinite-dimensional Cauchy problem.

31 pages

References in corpus (4)

• Orbital stability of Gausson solutions to logarithmic Schrödinger equations
• Existence of multi-solitons for the focusing Logarithmic Non-Linear Schrodinger Equation
• Logarithmic Schr{ö}dinger equation with quadratic potential
• Nonuniqueness and nonlinear instability of Gaussons under repulsive harmonic potential

Cited by in corpus (1)

• Sound propagation in cigar-shaped Bose liquids in the Thomas-Fermi approximation: A comparative study between Gross-Pitaevskii and logarithmic models