179 research outputs found
Krein-Space Formulation of PT-Symmetry, CPT-Inner Products, and Pseudo-Hermiticity
Emphasizing the physical constraints on the formulation of a quantum theory
based on the standard measurement axiom and the Schroedinger equation, we
comment on some conceptual issues arising in the formulation of PT-symmetric
quantum mechanics. In particular, we elaborate on the requirements of the
boundedness of the metric operator and the diagonalizability of the
Hamiltonian. We also provide an accessible account of a Krein-space derivation
of the CPT-inner product that was widely known to mathematicians since 1950's.
We show how this derivation is linked with the pseudo-Hermitian formulation of
PT-symmetric quantum mechanics.Comment: published version, 17 page
The Pauli equation with complex boundary conditions
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.Comment: 16 pages, 4 figure
PT-Symmetric Quantum Theory Defined in a Krein Space
We provide a mathematical framework for PT-symmetric quantum theory, which is
applicable irrespective of whether a system is defined on R or a complex
contour, whether PT symmetry is unbroken, and so on. The linear space in which
PT-symmetric quantum theory is naturally defined is a Krein space constructed
by introducing an indefinite metric into a Hilbert space composed of square
integrable complex functions in a complex contour. We show that in this Krein
space every PT-symmetric operator is P-Hermitian if and only if it has
transposition symmetry as well, from which the characteristic properties of the
PT-symmetric Hamiltonians found in the literature follow. Some possible ways to
construct physical theories are discussed within the restriction to the class
K(H).Comment: 8 pages, no figures; Refs. added, minor revisio
On elements of the Lax-Phillips scattering scheme for PT-symmetric operators
Generalized PT-symmetric operators acting an a Hilbert space
are defined and investigated. The case of PT-symmetric extensions of a
symmetric operator is investigated in detail. The possible application of
the Lax-Phillips scattering methods to the investigation of PT-symmetric
operators is illustrated by considering the case of 0-perturbed operators
-self-adjoint operators with -symmetries: extension theory approach
A well known tool in conventional (von Neumann) quantum mechanics is the
self-adjoint extension technique for symmetric operators. It is used, e.g., for
the construction of Dirac-Hermitian Hamiltonians with point-interaction
potentials. Here we reshape this technique to allow for the construction of
pseudo-Hermitian (-self-adjoint) Hamiltonians with complex
point-interactions. We demonstrate that the resulting Hamiltonians are
bijectively related with so called hypermaximal neutral subspaces of the defect
Krein space of the symmetric operator. This symmetric operator is allowed to
have arbitrary but equal deficiency indices . General properties of the
$\cC$ operators for these Hamiltonians are derived. A detailed study of
$\cC$-operator parametrizations and Krein type resolvent formulas is provided
for $J$-self-adjoint extensions of symmetric operators with deficiency indices
. The technique is exemplified on 1D pseudo-Hermitian Schr\"odinger and
Dirac Hamiltonians with complex point-interaction potentials
symmetric non-selfadjoint operators, diagonalizable and non-diagonalizable, with real discrete spectrum
Consider in , , the operator family . \ds
H_0= a^\ast_1a_1+... +a^\ast_da_d+d/2 is the quantum harmonic oscillator with
rational frequencies, a symmetric bounded potential, and a real
coupling constant. We show that if , being an explicitly
determined constant, the spectrum of is real and discrete. Moreover we
show that the operator \ds H(g)=a^\ast_1 a_1+a^\ast_2a_2+ig a^\ast_2a_1 has
real discrete spectrum but is not diagonalizable.Comment: 20 page
Conceptual Aspects of PT-Symmetry and Pseudo-Hermiticity: A status report
We survey some of the main conceptual developments in the study of
PT-symmetric and pseudo-Hermitian Hamiltonian operators that have taken place
during the past ten years or so. We offer a precise mathematical description of
a quantum system and its representations that allows us to describe the idea of
unitarization of a quantum system by modifying the inner product of the Hilbert
space. We discuss the role and importance of the quantum-to-classical
correspondence principle that provides the physical interpretation of the
observables in quantum mechanics. Finally, we address the problem of
constructing an underlying classical Hamiltonian for a unitary quantum system
defined by an a priori non-Hermitian Hamiltonian.Comment: 11 page
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