Quasi-exactly solvable quartic: elementary integrals and asymptotics

We study elementary eigenfunctions y = pe^h of operators L(y) = y'' + Py, where p, h and P are polynomials in one variable. For the case when h is an odd cubic polynomial, we investigate the real level crossing points and asymptotics of eigenvalues. This study leads to an interesting identity with elementary integrals.

Littlewood, when he makes use of an algebraic identity, always saves himself the trouble of proving it; he maintains that an identity, if true, can be verified in few lines by anybody obtuse enough to feel the need of verification.

Freeman Dyson 1949 Some guesses in theory of partitions Eureka 8 10–15