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Semiclassical expansion for exactly solvable differential operators

Jorge A. Borrego-Morell, Boris Shapiro

Abstract

Below we study a linear differential equation $\MM (v(z,η))=η^M{v(z,η)}$, where $η>0$ is a large spectral parameter and $\MM=\sum_{k=1}^{M}ρ_{k}(z)\frac{d^k}{dz^k},\; M\ge 2$ is a differential operator with polynomial coefficients such that the leading coefficient $ρ_M(z)$ is a monic complex-valued polynomial with $\dgr{ρ_M }=M$ and other $ρ_k(z)$'s are complex-valued polynomials with $\dgr{ρ_k }\leq k$. 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.

Semiclassical expansion for exactly solvable differential operators

Abstract

Below we study a linear differential equation , where is a large spectral parameter and is a differential operator with polynomial coefficients such that the leading coefficient is a monic complex-valued polynomial with and other 's are complex-valued polynomials with . We prove the Borel summability of its WKB-solutions in the Stokes regions. For under the assumption that 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.
Paper Structure (12 sections, 22 theorems, 163 equations, 9 figures)

This paper contains 12 sections, 22 theorems, 163 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Set-up of the problem
  3. Short historical account
  4. Preliminaries, basic notions, and formulation of main results
  5. Preliminaries
  6. Formulation of the main results
  7. Technical results
  8. Proofs of the main results
  9. Local structure of a third order exactly solvable differential operator near a turning point
  10. Euler-Cauchy equations
  11. Open problems
  12. Appendix. Sibuya's theorem on Gevrey summability of formal power series depending on a parameter.

Key Result

Theorem A

For a non–degenerate exactly solvable operator $\mathcal{M}$ of order $M \ge 2$ and the sequence $\{Q_n^\mathcal{M}(z)\}$ of its monic eigenpolynomials, when $n\to \infty$ one has the asymptotic expansion in the sense of Poincaré uniformly on compacts subsets $K\subset \Omega$, where for $j \ge 1$, $C_j$ are analytic in $\Omega$.

Figures (9)

  • Figure 1: Typical structure of $\hbox{supp}[\mu^\mathcal{M}]$ for $M=3$
  • Figure 2: Regions $V_{z_k}$ and $\Omega_Y$ in the $z$- and the $y$-spaces respectively.
  • Figure 3:
  • Figure 4: Integration paths for the Borel sums (wiggly lines denote the branch cuts for the Borel transforms).
  • Figure 5: The region $\hbox{supp}[\mu^\mathcal{M}]\setminus \mathfrak{r}$. The wiggly line denotes the branch cut defined by $\mathfrak{r}$
  • ...and 4 more figures

Theorems & Definitions (47)

  • Theorem A: see Theorem 1 of BoMo22
  • Definition 1
  • Example 1
  • Example 2
  • Definition 2
  • Remark 1
  • Definition 3
  • Remark 2
  • Definition 4: see Def. 1.2.1 p.21 H15vir, Ko00Fed93ka11
  • Remark 3
  • ...and 37 more