Cayley transform applied to non-interacting quantum transport

TL;DR

This paper extends the Landauer-Büttiker formalism to include unbounded operators and analyzes steady currents in complex quantum systems with semi-infinite leads and point interactions.

Contribution

It develops a generalized formalism for quantum transport involving unbounded operators and provides new formulas for steady currents in systems with semi-relativistic and Dirac operators.

Findings

Pure point and singular continuous subspaces do not contribute to steady current.

Derived stationary charge current formulas for systems with semi-infinite leads and dissipative boundary conditions.

Provided current formulas for electrons described by one-dimensional Dirac operators with point interactions.

Abstract

We extend the Landauer-B\"uttiker formalism in order to accommodate both unitary and self-adjoint operators which are not bounded from below. We also prove that the pure point and singular continuous subspaces of the decoupled Hamiltonian do not contribute to the steady current. One of the physical applications is a stationary charge current formula for a system with four pseudo-relativistic semi-infinite leads and with an inner sample which is described by a Schr\"odinger operator defined on a bounded interval with dissipative boundary conditions. Another application is a current formula for electrons described by a one dimensional Dirac operator; here the system consists of two semi-infinite leads coupled through a point interaction at zero.