D. Kochan, D. Krejcirik, R. Novak, P. Siegl
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.
http://arxiv.org/abs/1203.5011
Quantum Physics (quant-ph); High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph)
D. Krejcirik, P. Siegl, J. Zelezny
We consider one-dimensional Schroedinger-type operators in a bounded interval with non-self-adjoint Robin-type boundary conditions. It is well known that such operators are generically conjugate to normal operators via a similarity transformation. Motivated by recent interests in quasi-Hermitian Hamiltonians in quantum mechanics, we study properties of the transformations in detail. We show that they can be expressed as the sum of the identity and an integral Hilbert-Schmidt operator. In the case of parity and time reversal boundary conditions, we establish closed integral-type formulae for the similarity transformations, derive the similar self-adjoint operator and also find the associated “charge conjugation” operator, which plays the role of fundamental symmetry in a Krein-space reformulation of the problem.
http://arxiv.org/abs/1108.4946
Spectral Theory (math.SP); Mathematical Physics (math-ph); Functional Analysis (math.FA); Quantum Physics (quant-ph)
H. Hernandez-Coronado, D. Krejcirik, P. Siegl
![Transmissions |T|^2 as a function of energy k2 for the step-like potential <i>v</i> with a = Pi/4, epsilon_1 = 0.2, epsilon_3 = 0.5, beta_3 = -100, beta_2 = 0, beta_1 = -120 (continuous red line), and beta_1 = -200 (dashed blue line). See [5] for animated plots of |T|^2 as a function of potential.](http://ptsymmetry.net/wp-content/uploads/2010/11/figure4.png "figure4")
We establish that a perfect-transmission scattering problem can be described by a class of parity and time reversal symmetric operators and hereby we provide a scenario for understanding and implementing the corresponding quasi-Hermitian quantum mechanical framework from the physical viewpoint. One of the most interesting features of the analysis is that the complex eigenvalues of the underlying non-Hermitian problem, associated with a reflectionless scattering system, lead to the loss of perfect-transmission energies as the parameters characterizing the scattering potential are varied. On the other hand, the scattering data can serve to describe the spectrum of a large class of Schroedinger operators with complex Robin boundary conditions.
http://arxiv.org/abs/1011.4281
Mathematical Physics (math-ph); Quantum Physics (quant-ph)