Asymptotic Analysis - Volume 138, issue 3

Authors: Exner, Pavel | Kondej, Sylwia | Lotoreichik, Vladimir

Article Type: Research Article

Abstract: In this paper we consider the two-dimensional Schrödinger operator with an attractive potential which is a multiple of the characteristic function of an unbounded strip-shaped region, whose thickness is varying and is determined by the function R ∋ x ↦ d + ε f ( x ) , where d > 0 is a constant, ε > 0 is a small parameter, and f is a compactly supported continuous function. We prove that if ∫ R f d x > 0 , then the respective …Schrödinger operator has a unique simple eigenvalue below the threshold of the essential spectrum for all sufficiently small ε > 0 and we obtain the asymptotic expansion of this eigenvalue in the regime ε → 0 . An asymptotic expansion of the respective eigenfunction as ε → 0 is also obtained. In the case that ∫ R f d x < 0 we prove that the discrete spectrum is empty for all sufficiently small ε > 0 . In the critical case ∫ R f d x = 0 , we derive a sufficient condition for the existence of a unique bound state for all sufficiently small ε > 0 . Show more

Keywords: Schrödinger operators, strip-shaped potentials, discrete spectrum, weak deformation

DOI: 10.3233/ASY-241893

Citation: Asymptotic Analysis, vol. 138, no. 3, pp. 151-174, 2024

Authors: Barrué, Grégoire | Debussche, Arnaud | Tusseau, Maxime

Article Type: Research Article

Abstract: We prove that the stochastic Nonlinear Schrödinger (NLS) equation is the limit of NLS equation with random potential with vanishing correlation length. We generalize the perturbed test function method to the context of dispersive equations. Apart from the difficulty of working in infinite dimension, we treat the case of random perturbations which are not assumed uniformly bounded.

Keywords: Nonlinear Schrödinger equation, diffusion-approximation

DOI: 10.3233/ASY-241894

Citation: Asymptotic Analysis, vol. 138, no. 3, pp. 175-224, 2024