Asymptotics of the spectrum of a two-dimensional Hartree-type operator with Coulomb self-action potential near the lower boundaries of spectral clusters

Theoretical and Mathematical Physics. 2019. Vol. 199. No. 3. P. 864–877.

D. A. Vakhrameeva, A. V. Pereskokov

We consider the eigenvalue problem for a perturbed two-dimensional oscillator where the perturbation is an

integral Hartree-type nonlinearity with a Coulomb self-action potential. We obtain asymptotic eigenvalues

and asymptotic eigenfunctions near the lower boundaries of spectral clusters formed in a neighborhood of

the eigenvalues of the unperturbed operator and construct an asymptotic expansion near a circle where

the solution is localized.

We consider the eigenvalue problem for a perturbed two-dimensional oscillator where the perturbation is an

integral Hartree-type nonlinearity with a Coulomb self-action potential. We obtain asymptotic eigenvalues

and asymptotic eigenfunctions near the lower boundaries of spectral clusters formed in a neighborhood of

the eigenvalues of the unperturbed operator and construct an asymptotic expansion near a circle where

the solution is localized.

We consider the eigenvalue problem for a perturbed two-dimensional oscillator where the perturbation is an

integral Hartree-type nonlinearity with a Coulomb self-action potential. We obtain asymptotic eigenvalues

and asymptotic eigenfunctions near the lower boundaries of spectral clusters formed in a neighborhood of

the eigenvalues of the unperturbed operator and construct an asymptotic expansion near a circle where

 

the solution is localized.

We consider the eigenvalue problem for a perturbed two-dimensional oscillator where the perturbation is an

 

integral Hartree-type nonlinearity with a Coulomb self-action potential. We obtain asymptotic eigenvalues

 

and asymptotic eigenfunctions near the lower boundaries of spectral clusters formed in a neighborhood of

 

the eigenvalues of the unperturbed operator and construct an asymptotic expansion near a circle where

 

the solution is localized.

 

We consider the eigenvalue problem for a perturbed two-dimensional oscillator where the perturbation is an

integral Hartree-type nonlinearity with a Coulomb self-action potential. We obtain asymptotic eigenvalues

and asymptotic eigenfunctions near the lower boundaries of spectral clusters formed in a neighborhood of

the eigenvalues of the unperturbed operator and construct an asymptotic expansion near a circle where

the solution is localized.

We consider the eigenvalue problem for a perturbed two-dimensional oscillator where the perturbation is an

integral Hartree-type nonlinearity with a Coulomb self-action potential. We obtain asymptotic eigenvalues

and asymptotic eigenfunctions near the lower boundaries of spectral clusters formed in a neighborhood of

the eigenvalues of the unperturbed operator and construct an asymptotic expansion near a circle where

 

the solution is localized.

We consider the eigenvalue problem for a perturbed two-dimensional oscillator where the perturbation is an

 

integral Hartree-type nonlinearity with a Coulomb self-action potential. We obtain asymptotic eigenvalues

 

and asymptotic eigenfunctions near the lower boundaries of spectral clusters formed in a neighborhood of

 

the eigenvalues of the unperturbed operator and construct an asymptotic expansion near a circle where

 

the solution is localized.

 

We consider the eigenvalue problem for a perturbed two-dimensional oscillator where the perturbation is an

integral Hartree-type nonlinearity with a Coulomb self-action potential. We obtain asymptotic eigenvalues

and asymptotic eigenfunctions near the lower boundaries of spectral clusters formed in a neighborhood of

the eigenvalues of the unperturbed operator and construct an asymptotic expansion near a circle where

the solution is localized.

We consider the eigenvalue problem for a perturbed two-dimensional oscillator where the perturbation is an

integral Hartree-type nonlinearity with a Coulomb self-action potential. We obtain asymptotic eigenvalues

and asymptotic eigenfunctions near the lower boundaries of spectral clusters formed in a neighborhood of

the eigenvalues of the unperturbed operator and construct an asymptotic expansion near a circle where

the solution is localized.

We consider the eigenvalue problem for a perturbed two-dimensional oscillator where the perturbation is an

integral Hartree-type nonlinearity with a Coulomb self-action potential. We obtain asymptotic eigenvalues

and asymptotic eigenfunctions near the lower boundaries of spectral clusters formed in a neighborhood of

the eigenvalues of the unperturbed operator and construct an asymptotic expansion near a circle where

 

the solution is localized.

We consider the eigenvalue problem for a perturbed two-dimensional oscillator where the perturbation is an

 

integral Hartree-type nonlinearity with a Coulomb self-action potential. We obtain asymptotic eigenvalues

 

and asymptotic eigenfunctions near the lower boundaries of spectral clusters formed in a neighborhood of

 

the eigenvalues of the unperturbed operator and construct an asymptotic expansion near a circle where

 

the solution is localized.

 

We consider the eigenvalue problem for a perturbed two-dimensional oscillator where the perturbation is an

integral Hartree-type nonlinearity with a Coulomb self-action potential. We obtain asymptotic eigenvalues

and asymptotic eigenfunctions near the lower boundaries of spectral clusters formed in a neighborhood of

the eigenvalues of the unperturbed operator and construct an asymptotic expansion near a circle where

the solution is localized.

Keywords: спектральный кластер, asymptotic eigenvalues and eigenfunctions, асимптотические собственные значения и собственные функции, самосогласованное поле, self-consistent field, spectral cluster, spectrum splitting, расщепление спектра

Publication based on the results of: Mathematical Modeling of Resonance SystemsMathematical modeling of resonance systems (2019)