Borel Summable Solutions to 1D Schrödinger Equation

Abstract

It is shown that so called fundamental solutions the semiclassical expansions of which have been established earlier to be Borel summable to the solutions themselves appear also to be the unique solutions to the 1D Schrödinger equation having this property. Namely, it is shown in this paper that for the polynomial potentials the Borel function defined by the fundamental solutions can be considered as the canonical one. The latter means that any Borel summable solution can be obtained by the Borel transformation of this unique canonical Borel function multiplied by some $\hslash$-dependent and Borel summable constant. This justify the exceptional role the fundamental solutions play in 1D quantum mechanics and completes the relevant semiclassical theory relied on the Borel resummation technique and developed in our other papers.

39 pages, 20 Postscript figures, uses epsfig.sty article.sty, figure caption added and misprints corrected

References in corpus (2)

• Change of Variable as Borel Resummation of Semiclassical Series
• Fundamental solution method applied to time evolution of two energy level systems: exact and adiabatic limit results

Cited by in corpus (5)

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• Quantum anharmonic oscillators: a new approach
• Existence and Uniqueness of Exact WKB Solutions for Second-Order Singularly Perturbed Linear ODEs
• The high energy semiclassical asymptotics of loci of roots of fundamental solutions for polynomial potentials
• On the Borel summability of formal solutions of certain higher-order linear ordinary differential equations