ODE/IM correspondence in the semiclassical limit: Large degree asymptotics of the spectral determinants for the ground state potential

2024/09/12 by Degano, Gabriele
#Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2409.07866

Abstract

We study a Schrödinger-like equation for the anharmonic potential x^{2 α}+ℓ(ℓ+1) x^-2-E when the anharmonicity α goes to +∞. When E and ℓ vary in bounded domains, we show that the spectral determinant for the central connection problem converges to a special function written in terms of a Bessel function of order ℓ+(1)/(2) and its zeros converge to the zeros of that Bessel function. We then study the regime in which E and ℓ grow large as well, scaling as E∼ α² ε² and ℓ∼ αp. When ε is greater than 1 we show that the spectral determinant for the central connection problem is a rapidly oscillating function whose zeros tend to be distributed according to the continuous density law \frac2pπ(√(ε²-1))/(ε). When ε is close to 1 we show that the spectral determinant converges to a function expressed in terms of the Airy function Ai(-) and its zeros converge to the zeros of that function. This work is motivated by and has applications to the ODE/IM correspondence for the quantum KdV model.

Related

• Anharmonic oscillator, negative dimensions and inverse factorial convergence of large orders to the asymptotic form
• A primer of the complex WKB method, with application to the ODE/IM correspondence
• On the convergence of the usual perturbative expansions
• Long time growth of Sobolev norms in time dependent semiclassical anharmonic oscillators
• A new perturbation method in quantum mechanics