16 research outputs found
Characterization of potential smoothness and Riesz basis property of Hill-Scr\"odinger operators with singular periodic potentials in terms of periodic, antiperiodic and Neumann spectra
The Hill operators Ly=-y''+v(x)y, considered with singular complex valued
\pi-periodic potentials v of the form v=Q' with Q in L^2([0,\pi]), and subject
to periodic, antiperiodic or Neumann boundary conditions have discrete spectra.
For sufficiently large n, the disc {z: |z-n^2|<n} contains two periodic (if n
is even) or antiperiodic (if n is odd) eigenvalues \lambda_n^-, \lambda_n^+ and
one Neumann eigenvalue \nu_n. We show that rate of decay of the sequence
|\lambda_n^+-\lambda_n^-|+|\lambda_n^+ - \nu_n| determines the potential
smoothness, and there is a basis consisting of periodic (or antiperiodic) root
functions if and only if for even (respectively, odd) n, \sup_{\lambda_n^+\neq
\lambda_n^-}{|\lambda_n^+-\nu_n|/|\lambda_n^+-\lambda_n^-|} < \infty.Comment: arXiv admin note: substantial text overlap with arXiv:1207.094
Application of Pseudo-Hermitian Quantum Mechanics to a Complex Scattering Potential with Point Interactions
We present a generalization of the perturbative construction of the metric
operator for non-Hermitian Hamiltonians with more than one perturbation
parameter. We use this method to study the non-Hermitian scattering
Hamiltonian: H=p^2/2m+\zeta_-\delta(x+a)+\zeta_+\delta(x-a), where \zeta_\pm
and a are respectively complex and real parameters and \delta(x) is the Dirac
delta function. For regions in the space of coupling constants \zeta_\pm where
H is quasi-Hermitian and there are no complex bound states or spectral
singularities, we construct a (positive-definite) metric operator \eta and the
corresponding equivalent Hermitian Hamiltonian h. \eta turns out to be a
(perturbatively) bounded operator for the cases that the imaginary part of the
coupling constants have opposite sign, \Im(\zeta_+) = -\Im(\zeta_-). This in
particular contains the PT-symmetric case: \zeta_+ = \zeta_-^*. We also
calculate the energy expectation values for certain Gaussian wave packets to
study the nonlocal nature of \rh or equivalently the non-Hermitian nature of
\rH. We show that these physical quantities are not directly sensitive to the
presence of PT-symmetry.Comment: 22 pages, 4 figure
Physical Aspects of Pseudo-Hermitian and -Symmetric Quantum Mechanics
For a non-Hermitian Hamiltonian H possessing a real spectrum, we introduce a
canonical orthonormal basis in which a previously introduced unitary mapping of
H to a Hermitian Hamiltonian h takes a simple form. We use this basis to
construct the observables O of the quantum mechanics based on H. In particular,
we introduce pseudo-Hermitian position and momentum operators and a
pseudo-Hermitian quantization scheme that relates the latter to the ordinary
classical position and momentum observables. These allow us to address the
problem of determining the conserved probability density and the underlying
classical system for pseudo-Hermitian and in particular PT-symmetric quantum
systems. As a concrete example we construct the Hermitian Hamiltonian h, the
physical observables O, the localized states, and the conserved probability
density for the non-Hermitian PT-symmetric square well. We achieve this by
employing an appropriate perturbation scheme. For this system, we conduct a
comprehensive study of both the kinematical and dynamical effects of the
non-Hermiticity of the Hamiltonian on various physical quantities. In
particular, we show that these effects are quantum mechanical in nature and
diminish in the classical limit. Our results provide an objective assessment of
the physical aspects of PT-symmetric quantum mechanics and clarify its
relationship with both the conventional quantum mechanics and the classical
mechanics.Comment: 45 pages, 13 figures, 2 table