Tripartite connection condition for quantum graph vertex

TL;DR

This paper introduces a new $PQRS$-form for boundary conditions in quantum graph vertices, simplifying analysis and design of quantum filters, and improving upon the existing $ST$-form in certain applications.

Contribution

The paper presents the $PQRS$-form as a more effective boundary condition formulation for quantum graph vertices, enhancing analysis and filter design capabilities.

Findings

The $PQRS$-form reduces complexity in identifying independent parameters.

It simplifies the calculation of scattering matrices when matrices are singular.

The new form facilitates the design of generalized quantum filters.

Abstract

We discuss formulations of boundary conditions in a quantum graph vertex and demonstrate that the so-called $ST$-form can be further reduced up to a form more effective in certain applications: In particular, in identifying the number of independent parameters for given ranks of two connection matrices, or in calculating the scattering matrix when both matrices are singular. The new form of boundary conditions, called the $PQRS$-form, also gives a natural scheme to design generalized low and high pass quantum filters.