High energy asymptotics for the perturbed anharmonic oscillator

Abstract

In this article, we study the spectral properties of the perturbation of the generalized anharmonic oscillator. We consider a piecewise Hölder continuous perturbation and investigate how the Hölder constant can affect the eigenvalues. More precisely, we derive several first terms in the asymptotic expansion for the eigenvalues.

Keywords:

• High energy
• field theory
• perturbation
• quantum theory

AMS Subject Classifications:

• Primary: 34L15
• 34E15
• Secondary: 81Q10

Acknowledgments

The first author is grateful to the supervisor of her master's thesis, Vladimir Podolskii, as the article was inspired by the aforementioned master's thesis. We would like to thank Julie Rowlett for reading and commenting upon the preliminary version of this manuscript. We are also grateful to Grigori Rozenblum for the attention and useful comments and Simone Murro for helpful discussions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 We mean that there is a finite number of pieces such that V is Hölder continuous on each piece.

2 Note that the constants $C_0$,$C_1$, and $C_2$ are defined slightly differently than in [17].

3 We have chosen the notation in such a way that $f_{+}$ corresponds to a segment of a positive half-line, $[0,b]$, and $f_{-}$ corresponds to a segment of a negative half-line, $[-b,0]$.

Additional information

Funding

The research of the second author was partially supported by the grant of the Science Committee of Ministry of Education and Science of the Republic of Kazakhstan (grant AP08855579).