Semiclassical expansion for exactly solvable differential operators

2024/02/29 by Jorge A. Borrego-Morell, Boris Shapiro, Borrego-Morell, Jorge A. +1 · 1 citation
#math.CA

paper · pdf · doi:10.48550/arxiv.2402.19087

Abstract

Below we study a linear differential equation \MM (v(z,η))=η^Mv(z,η), where η>0 is a large spectral parameter and \MM=∑_k=1^Mρ_k(z)(d^k)/(dz^k), M≥ 2 is a differential operator with polynomial coefficients such that the leading coefficient ρ_M(z) is a monic complex-valued polynomial with \dgrρ_M =M and other ρ_k(z)'s are complex-valued polynomials with \dgrρ_k ≤ k. We prove the Borel summability of its WKB-solutions in the Stokes regions. For M=3 under the assumption that ρ_M has simple zeros, we give the full description of the Stokes complex (i.e. the union of all Stokes curves) of this equation. Finally, we show that for the Euler-Cauchy equations, their WKB-solutions converge in the usual sense.

Cited by

• Spectral networks for polynomial cubic differentials

Related

• Polynomial Solutions of Differential Equations
• Stokes Multipliers, Spectral Determinants and T-Q relations
• Resurgent Deformations for an Ordinary Differential Equation of Order 2
• A direct method to find Stokes multipliers in closed form for P1 and more general integrable systems
• A remark on 'Some numerical results in complex differential geometry'