The Schrödinger equation with a potential depending on a small parameter (Q1264785)

The paper is an extension of that published by the author previously [Teor. Mat. Fiz. 51, 54-72 (Russian. English summary) (1982; Zbl 0509.34022)]. The equation \(\varepsilon^2 Y''= P(z,\varepsilon) Y\) is considered; \((P(z,\varepsilon))\) is a polynomial of order \(n\) of the form \(P= z^n+ \alpha_1(\varepsilon) z^{n- 1}+ \alpha_2(\varepsilon) z^{n- 2}+\dots\), \(\varepsilon>0\) is a small parameter, \(\alpha_i(\varepsilon)\) are finite linear combinations of \(\varepsilon^{\mu_1},\varepsilon^{\mu_2},\dots, \mu_i>0\). Solutions to this equation are one-valued analytic functions of \(z\) on the whole plane and any three of them are linearly dependent. The problem is studied to determine these linear connections (Stokes constants). As an example the anharmonic oscillator is analyzed. The configuration of Stokes lines is discussed in the appendix.