Asymptotics and Monodromy of the Algebraic Spectrum of Quasi-Exactly Solvable Sextic Oscillator

ABSTRACT

In this article we study numerically and theoretically the asymptotics of the algebraic part of the spectrum for the quasi-exactly solvable sextic potential Π_{m, b}(x) = x⁶ + 2bx⁴ + (b² − (4m + 3))x², its level crossing points, and its monodromy in the complex plane of parameter b. Here m is a fixed positive integer. We also discuss the connection between the special sequence of quasi-exactly solvable sextics with increasing m and the classical quartic potential.

KEYWORDS:

• monodromy
• spectral surface
• spectrum of an harmonic oscillator

2000 AMS Subject Classification:

• 81Q10

Acknowledgments

The first author thanks Professors A. Eremenko and A. Gabrielov of Purdue University for numerous discussions. The first author is grateful to the Nuclear Physics Institute at Řež of the Czech Academy of Sciences for the hospitality in November 2016. The second author acknowledges the hospitality of the Department of Mathematics, Stockholm University in April 2016. Both authors are very grateful to the anonymous referee for the constructive criticism which allowed them to substantially improve the quality of the exposition.

Additional information

Funding

The second author’s research was supported by the Czech Science Foundation (GACR) within the project 14-06818S.