Semiclassical spectral series of a Hartree-type operator corresponding to a rest point of the classical Hamilton-Ehrenfest system

Abstract

We consider the classical equations of motion in quantum means, i.e., the Hamilton-Ehrenfest system. In the semiclassical approximation in the framework of the covariant approach based on these equations, we construct the spectral series of a nonlinear Hartree-type operator corresponding to a rest point.

Asymptotics of the Hartree operator spectrum near the upper boundaries of spectral clusters: Asymptotic solutions localized near a circle

“…are solutions of Eq. (20) in the space L 2 (R 2 ). They form a subspace H p ⊂ L 2 (R 2 ) whose orthonormal basis consists of the functions β j,p−j (τ, s), j = 0, 1, .…”

“…The theory of the semiclassical approximation of the Hartree equation and the asymptotic solutions of Hartree-type equations localized near small-dimensional invariant submanifolds in the phase space have been studied in numerous papers (see, e.g., [11]- [20]). Here, the leading term of the asymptotic expansion near the circle where the solution is localized is a solution of another classical problem in quantum mechanics, namely, the two-dimensional oscillator problem:…”

Symmetry and Intertwining Operators for the Nonlocal Gross-Pitaevskii Equation

“…where we used the notations The Hamilton-Ehrenfest system (2.15) for the first-order moments becomes [5] ṗ = −ρp − σ 0 x,…”

Quasi-energy spectral series for a nonlocal Gross-Pitaevskii equation

“…In the limit T → ∞ we can neglect the dependence of operators H(t) and V (t, Ψ) of the form (1.1) on time t. Therefore, the quasi-energy spectral series (5.16) grade into discrete spectral series of the nonlinear problem [ H + κ V (Ψ)]Ψ = EΨ in the limit T → ∞. These spectral series [28] are localized in a neighborhood of stationary solutions of the Hamilton-Ehrenfest system (3.2), (3.16). The geometrical phases (7.12) do not contain quantum corrections accurate to O( 1/2 ) in the considered example and do not depend on nonlinear potential.…”