Semiclassical Asymptotics of the Spectrum of the Hydrogen Atom in an Electromagnetic Field Near the Upper Boundaries of
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Journal of Mathematical Sciences, Vol. 251, No. 6, December, 2020
SEMICLASSICAL ASYMPTOTICS OF THE SPECTRUM OF THE HYDROGEN ATOM IN AN ELECTROMAGNETIC FIELD NEAR THE UPPER BOUNDARIES OF SPECTRAL CLUSTERS A. S. Migaeva National Research University “Moscow Power Engineering Institute” 14, Krasnokazarmennaya St., Moscow 111250, Russia anastasiy [email protected]
A. V. Pereskokov ∗ National Research University “Moscow Power Engineering Institute” 14, Krasnokazarmennaya St., Moscow 111250, Russia National Research University “Higher School of Economics” 20, Myasnitskaya St., Moscow 101978, Russia [email protected]
UDC 517.983
We study the Zeeman–Stark effect in the hydrogen atom located in an electromagnetic field by using irreducible representations of an algebra with the Karasev–Novikova quadratic commutation relations. The representations are associated with resonance spectral clusters near the energy level of the unperturbed hydrogen atom. We find asymptotics for a series of eigenvalues and corresponding asymptotic eigenfunctions near the upper boundaries of spectral clusters in the case of positive intensities of the electric field. Bibliography: 9 titles. Illustrations: 1 figure.
1
Introduction
We consider the nonrelativistic Hamiltonian of the hydrogen atom in a homogeneous electromagnetic field H = H0 + εM3 + εe1 x1 + ε2 W, (1.1) where H = −Δ − |x|−1 ,
M3 = ix2
∂ ∂ − ix1 , ∂x1 ∂x2
W = (x21 + x22 )/4.
Here, x = (x1 , x2 , x3 ) are the Cartesian coordinates in R3 , Δ is the Laplace operator, the magnetic field is directed along the x3 -axis, and the electric field is directed along the x1 -axis. The number e1 > 0 characterizes the electric field intensity, and ε > 0 is a small parameter. ∗
To whom the correspondence should be addressed.
Translated from Problemy Matematicheskogo Analiza 107, 2020, pp. 69-90. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2516-0850
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The problem of the hydrogen atom in an electromagnetic field is of great interest in physics and mathematics (cf., for example, [1]–[3]). The peculiarity of this problem is that an electric field and a magnetic field are simultaneously present in the Hamiltonian, and these fields are orthogonal to each other. This fact causes resonant spectral clusters near the eigenvalues of the unperturbed hydrogen atom [2, 4]. An algebraic method for constructing asymptotics of the spectrum and eigenfunctions of the hydrogen atom was proposed in [1]. The method is based on the quantum averaging of perturbation, the subsequent transition to the symmetry algebra, and the coherent transform from the original representation to the irreducible representation of the algebra in the space of functions over Lagrange submanifolds in a symplectic sheet. The method was later developed for problems with frequency resonances. Of particular interest are the states of the system described by the Hamiltonian (1.1) which correspond to the boundaries of spectral clusters. In this case, the above-mentioned Lagrange submanifolds almost collapse and it becomes impossible to r
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