Connection formula, hyperasymptotics, and Schrödinger eigenvalues: Dispersive hyperasymptotics and the anharmonic oscillator (Q2744089)

Let \(E_n(g)\) be the \(n\)th energy eigenvalue of the anharmonic oscillator whose Schrödinger equation is NEWLINE\[NEWLINE \Bigl(-\frac 12 \frac{d^2}{dx^2}+\frac{x^2}2+gx^4-E_n(g) \Bigr)\psi(x)=0. NEWLINE\]NEWLINE The function \(E_n(g)\) is expressible by an asymptotic power series in \(g\) and satisfies the once-subtracted dispersion relation NEWLINE\[NEWLINE E_n(g)=E^{(0)}_n+\frac{(-g)}{2\pi i}\int^{\infty}_0 \frac{\Delta E_n(z)}{z(z+g)} dz, \quad E^{(0)}_n=n+1/2, NEWLINE\]NEWLINE where \(\Delta E_n(z)\equiv E_n( e^{-i\pi}z)-E_n(e^{+i\pi}z)\) is the discontinuity of \(E_n(g)\) across the negative \(g\) axis when \(g=-z.\) Here, using this relation, the authors obtain a second-level hyperasymptotic expansion of \(E_n(g).\) Furthermore, errors for truncated partial sums are evaluated.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00055].