962 research outputs found
Quantum counterpart of spontaneously broken classical PT symmetry
The classical trajectories of a particle governed by the PT-symmetric
Hamiltonian () have been studied in
depth. It is known that almost all trajectories that begin at a classical
turning point oscillate periodically between this turning point and the
corresponding PT-symmetric turning point. It is also known that there are
regions in for which the periods of these orbits vary rapidly as
functions of and that in these regions there are isolated values of
for which the classical trajectories exhibit spontaneously broken PT
symmetry. The current paper examines the corresponding quantum-mechanical
systems. The eigenvalues of these quantum systems exhibit characteristic
behaviors that are correlated with those of the associated classical system.Comment: 11 pages, 7 figure
Complex WKB Analysis of a PT Symmetric Eigenvalue Problem
The spectra of a particular class of PT symmetric eigenvalue problems has
previously been studied, and found to have an extremely rich structure. In this
paper we present an explanation for these spectral properties in terms of
quantisation conditions obtained from the complex WKB method. In particular, we
consider the relation of the quantisation conditions to the reality and
positivity properties of the eigenvalues. The methods are also used to examine
further the pattern of eigenvalue degeneracies observed by Dorey et al. in
[1,2].Comment: 22 pages, 13 figures. Added references, minor revision
Calculation of the Hidden Symmetry Operator for a \cP\cT-Symmetric Square Well
It has been shown that a Hamiltonian with an unbroken \cP\cT symmetry also
possesses a hidden symmetry that is represented by the linear operator \cC.
This symmetry operator \cC guarantees that the Hamiltonian acts on a Hilbert
space with an inner product that is both positive definite and conserved in
time, thereby ensuring that the Hamiltonian can be used to define a unitary
theory of quantum mechanics. In this paper it is shown how to construct the
operator \cC for the \cP\cT-symmetric square well using perturbative
techniques.Comment: 10 pages, 2 figure
All Hermitian Hamiltonians Have Parity
It is shown that if a Hamiltonian is Hermitian, then there always exists
an operator P having the following properties: (i) P is linear and Hermitian;
(ii) P commutes with H; (iii) P^2=1; (iv) the nth eigenstate of H is also an
eigenstate of P with eigenvalue (-1)^n. Given these properties, it is
appropriate to refer to P as the parity operator and to say that H has parity
symmetry, even though P may not refer to spatial reflection. Thus, if the
Hamiltonian has the form H=p^2+V(x), where V(x) is real (so that H possesses
time-reversal symmetry), then it immediately follows that H has PT symmetry.
This shows that PT symmetry is a generalization of Hermiticity: All Hermitian
Hamiltonians of the form H=p^2+V(x) have PT symmetry, but not all PT-symmetric
Hamiltonians of this form are Hermitian
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
Identification of observables in quantum toboggans
Quantum systems with real energies generated by an apparently non-Hermitian
Hamiltonian may re-acquire the consistent probabilistic interpretation via an
ad hoc metric which specifies the set of observables in the updated Hilbert
space of states. The recipe is extended here to quantum toboggans. In the first
step the tobogganic integration path is rectified and the Schroedinger equation
is given the generalized eigenvalue-problem form. In the second step the
general double-series representation of the eligible metric operators is
derived.Comment: 25 p
A Class of Exactly-Solvable Eigenvalue Problems
The class of differential-equation eigenvalue problems
() on the interval
can be solved in closed form for all the eigenvalues and
the corresponding eigenfunctions . The eigenvalues are all integers and
the eigenfunctions are all confluent hypergeometric functions. The
eigenfunctions can be rewritten as products of polynomials and functions that
decay exponentially as . For odd the polynomials that are
obtained in this way are new and interesting classes of orthogonal polynomials.
For example, when N=1, the eigenfunctions are orthogonal polynomials in
multiplying Airy functions of . The properties of the polynomials for all
are described in detail.Comment: REVTeX, 16 pages, no figur
Does the complex deformation of the Riemann equation exhibit shocks?
The Riemann equation , which describes a one-dimensional
accelerationless perfect fluid, possesses solutions that typically develop
shocks in a finite time. This equation is \cP\cT symmetric. A one-parameter
\cP\cT-invariant complex deformation of this equation,
( real), is solved exactly using the
method of characteristic strips, and it is shown that for real initial
conditions, shocks cannot develop unless is an odd integer.Comment: latex, 8 page
Semiclassical analysis of a complex quartic Hamiltonian
It is necessary to calculate the C operator for the non-Hermitian
PT-symmetric Hamiltonian H=\half p^2+\half\mu^2x^2-\lambda x^4 in order to
demonstrate that H defines a consistent unitary theory of quantum mechanics.
However, the C operator cannot be obtained by using perturbative methods.
Including a small imaginary cubic term gives the Hamiltonian H=\half p^2+\half
\mu^2x^2+igx^3-\lambda x^4, whose C operator can be obtained perturbatively. In
the semiclassical limit all terms in the perturbation series can be calculated
in closed form and the perturbation series can be summed exactly. The result is
a closed-form expression for C having a nontrivial dependence on the dynamical
variables x and p and on the parameter \lambda.Comment: 4 page
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