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Monodromy matrices. Stokes multipliers and harmonic oscillator. (Romanian. English summary) Zbl 0617.34002

This paper is a review of the main results concerning the characterization of solutions of second order linear differential equations in the complex domain, in the neighbourhood of regular and irregular singular points. The monodromy matrix at singular point \(z_ k\) \((k=1,2,...,n)\) tells us how a pair of linear independent solutions \(v_ j\) \((j=1,2)\) changes when z moves along a path around \(z_ k\). For an irregular singularity, we must study asymptotic expansions \(t_ j\) \((j=1,2)\) of solutions in certain sectors \(S_{\ell}\) \((\ell =1,2,...,N)\) of the complex plane. One can determine N canonical bases \(\{v_{1^{\ell}},v_{2^{\ell}}\}_{\ell =1,2,...,N}\) with the properties that \(v_{j,\ell}\sim t_ j\) in \(S_{\ell}\), and \(\{v_{1^{\ell}},v_{2^{\ell}}\}\) is related to \(\{v_{1,\ell - 1},v_{2,\ell -1}\}\) or \(\{v_{1,\ell +1},v_{2,\ell +1}\}\) by the Stokes multipliers (2\(\times 2\) matrices of the form \(\left( \begin{matrix} 1\\ *\end{matrix} \begin{matrix} 0\\ 1\end{matrix} \right)\) or \(\left( \begin{matrix} 1\\ 0\end{matrix} \begin{matrix} *\\ 1\end{matrix} \right))\). There exists a relation between Stokes multipliers, the monodromy matrix of solutions and the formal monodromy matrix of asymptotic expansions. In the case of the Schrödinger equation for the harmonic oscillator, the Stokes multipliers can be found directly from this relation; they depend on the system’s energy only. A ”good” solution must vanish when \(x=Re(z)\to \pm \infty\). Thus Stokes multipliers must satisfy a certain condition, which determines the quantization of energy levels.

MSC:

34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
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