On characteristic functions of operators on equilateral graphs. (English) Zbl 1265.34114

Pivovarchik, V.; Taystruk, O.

Methods Funct. Anal. Topol. 18, No. 2, 189-197 (2012).

A “quantum graph” is a selfadjoint operator determined by the Sturm-Liouville differential equation with appropriate boundary and matching conditions on a finite metric graph. Its characteristic function is an entire function whose zeroes are the eigenvalues; see [C. K. Law and V. Pivovarchik, J. Phys. A, Math. Theor. 42, No. 3, Article ID 035302 (2009; Zbl 1171.34017)]. The authors use the known connection between discrete and continuous Laplacians with the same symmetric potential on graphs to construct the characteristic functions. Relations between their properties and the geometry of graphs are investigated.

Reviewer: A. N. Kochubei (Kyïv)

MSC:

34B45	Boundary value problems on graphs and networks for ordinary differential equations
34B24	Sturm-Liouville theory
34A55	Inverse problems involving ordinary differential equations
34L15	Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators

Keywords:

quantum graph; Sturm-Liouville equation; discrete Laplacian; characteristic function

Citations:

Zbl 1171.34017