Approximation of quantum graph vertex couplings by scaled Schrödinger operators on thin branched manifolds

Abstract

Abstract. We discuss approximations of vertex couplings of quantum graphs using families of thin branched manifolds. We show that if a Neumann type Laplacian on such manifolds is amended by suitable potentials, the resulting Schrödinger operators can approximate non-trivial vertex couplings. The latter include not only the δ-couplings but also those with wavefunctions discontinuous at the vertex. We work out the example of the symmetric δ ′ -couplings and conjecture that the same method can be applied to all couplings invariant with respect to the time reversal.

New isotope technologies in environmental physics

“…This "fat" graph is some neighbourhood of a metric graph and, of course, corresponding partial differential equations should be solved to find its spectral and resonance properties. For results on convergence of "fat" graphs to quantum graphs we refer to the book [Pos12] or a series of papers by Exner and Post [EP05,EP07,EP09,EP13]. These results show that quantum graphs are feasible approximations of graph-like structures made from quantum wires.…”

On vertex conditions for elastic systems

“…One may prove that these conditions appear when approximating of beams by narrow channels is carried out similar to [EP,EP1,EP2,G,GJ,KZ,KZ1]. It is straightforward to generalize obtained conditions to the case where the number of beams is greater than 3.…”

Nonlinear standing waves on planar branched systems: shrinking into metric graph

“…In particular, stationary linear wave equations on fat graphs have been studied in the Refs. [11][12][13][26][27][28] (see [29,30] and references therein for detailed reviews). The corresponding nonlinear problem has been mainly studied for the one-dimensional case by considering the metric graph approach Different aspects of the nonlinear Schrödinger equation for branched one dimensional branched domains called metric graphs were previously studied earlier [4][5][6][7][8][9][10].…”

“…Such systems can be modeled by so-called fat graphs. Previously, the linear Schrödinger equation on fat graphs was addressed in a number of works [11][12][13] by considering metric graph limit as transition to from planar to linear wave motions. Extension of such a study to the case of nonlinear Schrödinger equation based on the numerical treatment of the problem was done in recent work [14].…”