9,035 research outputs found

### WKB Analysis of PT-Symmetric Sturm-Liouville problems

Most studies of PT-symmetric quantum-mechanical Hamiltonians have considered
the Schroedinger eigenvalue problem on an infinite domain. This paper examines
the consequences of imposing the boundary conditions on a finite domain. As is
the case with regular Hermitian Sturm-Liouville problems, the eigenvalues of
the PT-symmetric Sturm-Liouville problem grow like $n^2$ for large $n$.
However, the novelty is that a PT eigenvalue problem on a finite domain
typically exhibits a sequence of critical points at which pairs of eigenvalues
cease to be real and become complex conjugates of one another. For the
potentials considered here this sequence of critical points is associated with
a turning point on the imaginary axis in the complex plane. WKB analysis is
used to calculate the asymptotic behaviors of the real eigenvalues and the
locations of the critical points. The method turns out to be surprisingly
accurate even at low energies.Comment: 11 pages, 8 figure

### Quantum tunneling as a classical anomaly

Classical mechanics is a singular theory in that real-energy classical
particles can never enter classically forbidden regions. However, if one
regulates classical mechanics by allowing the energy E of a particle to be
complex, the particle exhibits quantum-like behavior: Complex-energy classical
particles can travel between classically allowed regions separated by potential
barriers. When Im(E) -> 0, the classical tunneling probabilities persist.
Hence, one can interpret quantum tunneling as an anomaly. A numerical
comparison of complex classical tunneling probabilities with quantum tunneling
probabilities leads to the conjecture that as ReE increases, complex classical
tunneling probabilities approach the corresponding quantum probabilities. Thus,
this work attempts to generalize the Bohr correspondence principle from
classically allowed to classically forbidden regions.Comment: 12 pages, 7 figure

### Harmonic oscillator well with a screened Coulombic core is quasi-exactly solvable

In the quantization scheme which weakens the hermiticity of a Hamiltonian to
its mere PT invariance the superposition V(x) = x^2+ Ze^2/x of the harmonic and
Coulomb potentials is defined at the purely imaginary effective charges
(Ze^2=if) and regularized by a purely imaginary shift of x. This model is
quasi-exactly solvable: We show that at each excited, (N+1)-st
harmonic-oscillator energy E=2N+3 there exists not only the well known harmonic
oscillator bound state (at the vanishing charge f=0) but also a normalizable
(N+1)-plet of the further elementary Sturmian eigenstates \psi_n(x) at
eigencharges f=f_n > 0, n = 0, 1, ..., N. Beyond the first few smallest
multiplicities N we recommend their perturbative construction.Comment: 13 pages, Latex file, to appear in J. Phys. A: Math. Ge

### PT-symmetric sextic potentials

The family of complex PT-symmetric sextic potentials is studied to show that
for various cases the system is essentially quasi-solvable and possesses real,
discrete energy eigenvalues. For a particular choice of parameters, we find
that under supersymmetric transformations the underlying potential picks up a
reflectionless part.Comment: 8 pages, LaTeX with amssym, no figure

### Complex Extension of Quantum Mechanics

It is shown that the standard formulation of quantum mechanics in terms of
Hermitian Hamiltonians is overly restrictive. A consistent physical theory of
quantum mechanics can be built on a complex Hamiltonian that is not Hermitian
but satisfies the less restrictive and more physical condition of space-time
reflection symmetry (PT symmetry). Thus, there are infinitely many new
Hamiltonians that one can construct to explain experimental data. One might
expect that a quantum theory based on a non-Hermitian Hamiltonian would violate
unitarity. However, if PT symmetry is not spontaneously broken, it is possible
to construct a previously unnoticed physical symmetry C of the Hamiltonian.
Using C, an inner product is constructed whose associated norm is positive
definite. This construction is completely general and works for any
PT-symmetric Hamiltonian. Observables exhibit CPT symmetry, and the dynamics is
governed by unitary time evolution. This work is not in conflict with
conventional quantum mechanics but is rather a complex generalisation of it.Comment: 4 Pages, Version to appear in PR

### 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

### Scalar Quantum Field Theory with Cubic Interaction

In this paper it is shown that an i phi^3 field theory is a physically
acceptable field theory model (the spectrum is positive and the theory is
unitary). The demonstration rests on the perturbative construction of a linear
operator C, which is needed to define the Hilbert space inner product. The C
operator is a new, time-independent observable in PT-symmetric quantum field
theory.Comment: Corrected expressions in equations (20) and (21

### Exact PT-Symmetry Is Equivalent to Hermiticity

We show that a quantum system possessing an exact antilinear symmetry, in
particular PT-symmetry, is equivalent to a quantum system having a Hermitian
Hamiltonian. We construct the unitary operator relating an arbitrary
non-Hermitian Hamiltonian with exact PT-symmetry to a Hermitian Hamiltonian. We
apply our general results to PT-symmetry in finite-dimensions and give the
explicit form of the above-mentioned unitary operator and Hermitian Hamiltonian
in two dimensions. Our findings lead to the conjecture that non-Hermitian
CPT-symmetric field theories are equivalent to certain nonlocal Hermitian field
theories.Comment: Few typos have been corrected and a reference update

### An SCF-stabilization approach to excited states embedded in the continuum

By using SCF and stabilizationâlike procedures, we have located a (Ï, Ï*) singlet resonanceâlike state in ethylene at 10.21 eV. This state is embedded in the ionization continuum and carries an oscillator strength of 0.46 and is probably the analog to the spectroscopic V state in HartreeâFock theory. Implications of these results for other systems are discussed

### Quantum counterpart of spontaneously broken classical PT symmetry

The classical trajectories of a particle governed by the PT-symmetric
Hamiltonian $H=p^2+x^2(ix)^\epsilon$ ($\epsilon\geq0$) 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 $\epsilon$ for which the periods of these orbits vary rapidly as
functions of $\epsilon$ and that in these regions there are isolated values of
$\epsilon$ 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

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