282 research outputs found
Delta-Function Potential with a Complex Coupling
We explore the Hamiltonian operator H=-d^2/dx^2 + z \delta(x) where x is
real, \delta(x) is the Dirac delta function, and z is an arbitrary complex
coupling constant. For a purely imaginary z, H has a (real) spectral
singularity at E=-z^2/4. For \Re(z)<0, H has an eigenvalue at E=-z^2/4. For the
case that \Re(z)>0, H has a real, positive, continuous spectrum that is free
from spectral singularities. For this latter case, we construct an associated
biorthonormal system and use it to perform a perturbative calculation of a
positive-definite inner product that renders H self-adjoint. This allows us to
address the intriguing question of the nonlocal aspects of the equivalent
Hermitian Hamiltonian for the system. In particular, we compute the energy
expectation values for various Gaussian wave packets to show that the
non-Hermiticity effect diminishes rapidly outside an effective interaction
region.Comment: Published version, 14 pages, 2 figure
Pseudo-Unitary Operators and Pseudo-Unitary Quantum Dynamics
We consider pseudo-unitary quantum systems and discuss various properties of
pseudo-unitary operators. In particular we prove a characterization theorem for
block-diagonalizable pseudo-unitary operators with finite-dimensional diagonal
blocks. Furthermore, we show that every pseudo-unitary matrix is the
exponential of times a pseudo-Hermitian matrix, and determine the
structure of the Lie groups consisting of pseudo-unitary matrices. In
particular, we present a thorough treatment of pseudo-unitary
matrices and discuss an example of a quantum system with a
pseudo-unitary dynamical group. As other applications of our general results we
give a proof of the spectral theorem for symplectic transformations of
classical mechanics, demonstrate the coincidence of the symplectic group
with the real subgroup of a matrix group that is isomorphic to the
pseudo-unitary group U(n,n), and elaborate on an approach to second
quantization that makes use of the underlying pseudo-unitary dynamical groups.Comment: Revised and expanded version, includes an application to symplectic
transformations and groups, accepted for publication in J. Math. Phy
Coherent states of non-Hermitian quantum systems
We use the Gazeau-Klauder formalism to construct coherent states of
non-Hermitian quantum systems. In particular we use this formalism to construct
coherent state of a PT symmetric system. We also discuss construction of
coherent states following Klauder's minimal prescription.Comment: to appear in Phys.Lett
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
Construction of a unique metric in quasi-Hermitian quantum mechanics: non-existence of the charge operator in a 2 x 2 matrix model
For a specific exactly solvable 2 by 2 matrix model with a PT-symmetric
Hamiltonian possessing a real spectrum, we construct all the eligible physical
metrics and show that none of them admits a factorization CP in terms of an
involutive charge operator C. Alternative ways of restricting the physical
metric to a unique form are briefly discussed.Comment: 13 page
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
Path-Integral Formulation of Pseudo-Hermitian Quantum Mechanics and the Role of the Metric Operator
We provide a careful analysis of the generating functional in the path
integral formulation of pseudo-Hermitian and in particular PT-symmetric quantum
mechanics and show how the metric operator enters the expression for the
generating functional.Comment: Published version, 4 page
Interactions of Hermitian and non-Hermitian Hamiltonians
The coupling of non-Hermitian PT-symmetric Hamiltonians to standard Hermitian
Hamiltonians, each of which individually has a real energy spectrum, is
explored by means of a number of soluble models. It is found that in all cases
the energy remains real for small values of the coupling constant, but becomes
complex if the coupling becomes stronger than some critical value. For a
quadratic non-Hermitian PT-symmetric Hamiltonian coupled to an arbitrary real
Hermitian PT-symmetric Hamiltonian, the reality of the ground-state energy for
small enough coupling constant is established up to second order in
perturbation theory.Comment: 9 pages, 0 figure
On Pseudo-Hermitian Hamiltonians and Their Hermitian Counterparts
In the context of two particularly interesting non-Hermitian models in
quantum mechanics we explore the relationship between the original Hamiltonian
H and its Hermitian counterpart h, obtained from H by a similarity
transformation, as pointed out by Mostafazadeh. In the first model, due to
Swanson, h turns out to be just a scaled harmonic oscillator, which explains
the form of its spectrum. However, the transformation is not unique, which also
means that the observables of the original theory are not uniquely determined
by H alone. The second model we consider is the original PT-invariant
Hamiltonian, with potential V=igx^3. In this case the corresponding h, which we
are only able to construct in perturbation theory, corresponds to a complicated
velocity-dependent potential. We again explore the relationship between the
canonical variables x and p and the observables X and P.Comment: 9 pages, no figure
Non-Hermitian Hamiltonians with real and complex eigenvalues in a Lie-algebraic framework
We show that complex Lie algebras (in particular sl(2,C)) provide us with an
elegant method for studying the transition from real to complex eigenvalues of
a class of non-Hermitian Hamiltonians: complexified Scarf II, generalized
P\"oschl-Teller, and Morse. The characterizations of these Hamiltonians under
the so-called pseudo-Hermiticity are also discussed.Comment: LaTeX, 14 pages, no figure, 1 reference adde
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