Nonclassical Sturm-Liouville problems and Schrödinger operators on radial trees. (English) Zbl 0960.34070
Summary: Schrödinger operators on graphs with weighted edges may be defined using possibly infinite systems of ordinary differential operators. This work mainly considers radial trees, whose branching and edge lengths depend only on the distance from the root vertex. The analysis of operators with radial coefficients on radial trees is reduced, by a method analogous to separation of variables, to nonclassical boundary value problems on the line with interior point conditions. This reduction is used to study selfadjoint problems requiring boundary conditions ‘at infinity’.
MSC:
34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |
34B24 | Sturm-Liouville theory |
34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |
47E05 | General theory of ordinary differential operators |
34A35 | Ordinary differential equations of infinite order |