Asymptotic solution of stationary nonlinear Hartree equation

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

“…We invert the matrix JM κ (z 0 * ) and hence easily obtain the first-order correction z 1 * in the z component of the rest point from Eq. (23). As a result, we obtain…”

“…The solution of spectral problem (40) for k = 0 and H = 0 with a Gaussian-type interaction potential is given in [23].…”

Semiclassical spectral series for the two-component Hartree-type equation

“…At 0 = ae , all obtained constructions are transformed into solutions for a linear problem [35] (see also [38]). The results obtained allow the quantum adiabatic evolution of the two-component system to be investigated for slow variations of the system parameters (see [32]).…”

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

“…We further expand (1+(n r +1)/ ) −2 in a power series in and substitute asymptotic forms (16) and (17) in (13) and (14). It follows from formulas (12), (17), (19), and (14) that it suffices to set E 0 = 1/4π for the terms of the order of −2 log to be absent in the left-hand side of (13).…”

“…We use the Taylor formula to expand the functions in (13) and (14) in power series in τ and s. Because the function K(κ) has a logarithmic singularity as κ → 1 [26],…”