Semiclassical Asymptotics of the Spectrum of a Two-Dimensional Hartree Type Operator Near Boundaries of Spectral Clusters

Pereskokov, A. V.

doi:10.1007/s10958-022-06021-5

Cited by 1 publication

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“…The semiclassical approximation is widely used for linear equations of quantum mechanics. Some modern semiclassical approaches based on ideas of the Maslov method [33] were also applied to some nonlinear problems (see, e.g., [34,35]). In [36][37][38], the formalism of semiclassical asymptotics for a generalized nonlocal GPE in a special class of trajectory concentrated functions is developed that corresponds to closed quantum systems.…”

Section: Introductionmentioning

confidence: 99%

A Semiclassical Approach to the Nonlocal Nonlinear Schrödinger Equation with a Non-Hermitian Term

Kulagin,

Shapovalov

2024

Mathematics

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The nonlinear Schrödinger equation (NLSE) with a non-Hermitian term is the model for various phenomena in nonlinear open quantum systems. We deal with the Cauchy problem for the nonlocal generalization of multidimensional NLSE with a non-Hermitian term. Using the ideas of the Maslov method, we propose the method of constructing asymptotic solutions to this equation within the framework of semiclassically concentrated states. The semiclassical nonlinear evolution operator and symmetry operators for the leading term of asymptotics are derived. Our approach is based on the solutions of the auxiliary dynamical system that effectively linearizes the problem under certain algebraic conditions. The formalism proposed is illustrated with the specific example of the NLSE with a non-Hermitian term that is the model of an atom laser. The analytical asymptotic solution to the Cauchy problem is obtained explicitly for this example.

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