The Pauli equation with complex boundary conditions

Abstract

We consider one-dimensional Pauli Hamiltonians in a bounded interval with possibly non-self-adjoint Robin-type boundary conditions. We study the influence of the spin-magnetic interaction on the interplay between the type of boundary conditions and the spectrum. A special attention is paid to PT-symmetric boundary conditions with the physical choice of the time-reversal operator T.

A simple criterion for the existence of nonreal eigenvalues for a class of 2D and 3D Pauli operators

“…The author gratefully acknowledges the many helpful suggestions of V. Bruneau during the preparation of the paper. The author should like to thank R. Novák for bringing to his attention the reference [24].…”

Mathematical and physical aspects of complex symmetric operators

“…Pauli equation with complex boundary conditions. An interesting example of a PT -symmetric spin- 1 2 system is the Pauli Hamiltonian [94]:…”

“…Reference [94] analyzed the model in some simplifying circumstances, namely, for B = (0, 0, B) in which case B • L and (B × x) 2 act only on the first two coordinates and B • σ reduces to Bσ 3 . The domain was taken to be Ω = R 2 × (−a, a) and the matrix A entering the boundary condition was taken independent of the first two space-coordinates.…”

Non-self-adjoint graphs

“…We remark that the time-reversion in quantum mechanics can be more complicated in spinorial models and it can be non-involutive, cf. for instance [23], where Pauli equation is discussed. The origin of non-involutivity is the nontrivial action of J on the spinor components.…”