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

In this article we study numerically and theoretically the asymptotics of the algebraic part of the spec- trum for the quasi-exactly solvable sextic potential 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 xed positive integer. We also discuss the connection between the special sequence of quasi-exactly solvable sextics with increasing m and the classical quartic potential.