The regular cyclic matrix of an isolated singular point of the Sturm--Liouville equation of the standard form

A. A. Golubkov

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh international spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings—XXXIV", Voronezh, May 3-9, 2023, Part 4, Tome 233 (2024), pp. 3-13.

Résumé

For the Sturm–Liouville equation of the standard form, we examine properties of the transfer matrix $\hat{C}$ along a closed path starting at a point $z_0$ and going counterclockwise around the boundary of a convex domain containing exactly one singular point $z_s$ of the potential (the boundary of the domain does not contain singular points). The main attention is paid to the study of singular points that are not branching points; we prove that in this case, if the trace of the matrix $\hat{C}$ is not equal to two, then all its elements are entire functions of the spectral parameter of order $1/2$ and type $2|z_0 - z_s|$ with a trigonometric indicator.

Détail

Mots-clés : Sturm–Liouville equations on the complex plane, transfer matrix
Keywords: singular points

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