WIAS Preprint No. 1876, (2013)

The Cayley transform applied to non-interacting quantum transport


  • Cornean, Horia
  • Neidhardt, Hagen
  • Wilhelm, Lukas
  • Zagrebnov, Valentin

2010 Mathematics Subject Classification

  • 47A40 47A55 81Q37 81V80


  • Landauer-Büttiker formula, dissipative Schrödinger operators, self-adjoint dilations, Dirac operators




We extend the Landauer-Büttiker 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 Schrödinger 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.

Appeared in

  • J. Funct. Anal., 266 (2014) pp. 1421--1475.

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