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 spectrum for the quasi-exactly solvable sextic potential pi(m, b)(x) = x(6) + 2bx(4) + (b(2) - (4m + 3))x(2), 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.