11,825 research outputs found
Multiple-Scale Analysis of the Quantum Anharmonic Oscillator
Conventional weak-coupling perturbation theory suffers from problems that
arise from resonant coupling of successive orders in the perturbation series.
Multiple-scale perturbation theory avoids such problems by implicitly
performing an infinite reordering and resummation of the conventional
perturbation series. Multiple-scale analysis provides a good description of the
classical anharmonic oscillator. Here, it is extended to study the Heisenberg
operator equations of motion for the quantum anharmonic oscillator. The
analysis yields a system of nonlinear operator differential equations, which is
solved exactly. The solution provides an operator mass renormalization of the
theory.Comment: 12 pages, Revtex, no figures, available through anonymous ftp from
ftp://euclid.tp.ph.ic.ac.uk/papers/ or on WWW at
http://euclid.tp.ph.ic.ac.uk/Papers/papers_95-6_.htm
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
On the eigenproblems of PT-symmetric oscillators
We consider the non-Hermitian Hamiltonian H=
-\frac{d^2}{dx^2}+P(x^2)-(ix)^{2n+1} on the real line, where P(x) is a
polynomial of degree at most n \geq 1 with all nonnegative real coefficients
(possibly P\equiv 0). It is proved that the eigenvalues \lambda must be in the
sector | arg \lambda | \leq \frac{\pi}{2n+3}. Also for the case
H=-\frac{d^2}{dx^2}-(ix)^3, we establish a zero-free region of the
eigenfunction u and its derivative u^\prime and we find some other interesting
properties of eigenfunctions.Comment: 21pages, 9 figure
Model of supersymmetric quantum field theory with broken parity symmetry
Recently, it was observed that self-interacting scalar quantum field theories
having a non-Hermitian interaction term of the form ,
where is a real positive parameter, are physically acceptable in the
sense that the energy spectrum is real and bounded below. Such theories possess
PT invariance, but they are not symmetric under parity reflection or time
reversal separately. This broken parity symmetry is manifested in a nonzero
value for , even if is an even integer. This paper extends
this idea to a two-dimensional supersymmetric quantum field theory whose
superpotential is . The resulting quantum
field theory exhibits a broken parity symmetry for all . However,
supersymmetry remains unbroken, which is verified by showing that the
ground-state energy density vanishes and that the fermion-boson mass ratio is
unity.Comment: 20 pages, REVTeX, 11 postscript figure
Nonlinear Integral-Equation Formulation of Orthogonal Polynomials
The nonlinear integral equation P(x)=\int_alpha^beta dy w(y) P(y) P(x+y) is
investigated. It is shown that for a given function w(x) the equation admits an
infinite set of polynomial solutions P(x). For polynomial solutions, this
nonlinear integral equation reduces to a finite set of coupled linear algebraic
equations for the coefficients of the polynomials. Interestingly, the set of
polynomial solutions is orthogonal with respect to the measure x w(x). The
nonlinear integral equation can be used to specify all orthogonal polynomials
in a simple and compact way. This integral equation provides a natural vehicle
for extending the theory of orthogonal polynomials into the complex domain.
Generalizations of the integral equation are discussed.Comment: 7 pages, result generalized to include integration in the complex
domai
PT Symmetry as a Generalization of Hermiticity
The Hilbert space in PT-symmetric quantum mechanics is formulated as a linear
vector space with a dynamic inner product. The most general PT-symmetric matrix
Hamiltonians are constructed for 2*2 and 3*3 cases. In the former case, the
PT-symmetric Hamiltonian represents the most general matrix Hamiltonian with a
real spectrum. In both cases, Hermitian matrices are shown to be special cases
of PT-symmetric matrices. This finding confirms and strengthens the early
belief that the PT-symmetric quantum mechanics is a generalization of the
conventional Hermitian quantum mechanics.Comment: 13 page
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
Light from Cosmic Strings
The time-dependent metric of a cosmic string leads to an effective
interaction between the string and photons - the "gravitational Aharonov-Bohm"
effect -- and causes cosmic strings to emit light. We evaluate the radiation of
pairs of photons from cosmic strings and find that the emission from cusps,
kinks and kink-kink collisions occurs with a flat spectrum at all frequencies
up to the string scale. Further, cusps emit a beam of photons, kinks emit along
a curve, and the emission at a kink-kink collision is in all directions. The
emission of light from cosmic strings could provide an important new
observational signature of cosmic strings that is within reach of current
experiments for a range of string tensions.Comment: 8 pages, 1 figur
PT-symmetry and its spontaneous breakdown explained by anti-linearity
The impact of an anti-unitary symmetry on the spectrum of non-Hermitian operators is studied. Wigner's normal form of an anti-unitary operator accounts for the spectral properties of non-Hermitian, PE-symmetric Harniltonians. The occurrence of either single real or complex conjugate pairs of eigenvalues follows from this theory. The corresponding energy eigenstates span either one- or two-dimensional irreducible representations of the symmetry PE. In this framework, the concept of a spontaneously broken PE-symmetry is not needed
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