10 research outputs found
Some properties of eigenvalues and eigenfunctions of the cubic oscillator with imaginary coupling constant
Comparison between the exact value of the spectral zeta function,
, and the results
of numeric and WKB calculations supports the conjecture by Bessis that all the
eigenvalues of this PT-invariant hamiltonian are real. For one-dimensional
Schr\"odinger operators with complex potentials having a monotonic imaginary
part, the eigenfunctions (and the imaginary parts of their logarithmic
derivatives) have no real zeros.Comment: 6 pages, submitted to J. Phys.
On integrable Hamiltonians for higher spin XXZ chain
Integrable Hamiltonians for higher spin periodic XXZ chains are constructed
in terms of the spin generators; explicit examples for spins up to 3/2 are
given. Relations between Hamiltonians for some U_q(sl_2)-symmetric and
U(1)-symmetric universal r-matrices are studied; their properties are
investigated. A certain modification of the higher spin periodic chain
Hamiltonian is shown to be an integrable U_q(sl_2)-symmetric Hamiltonian for an
open chain.Comment: 20 pages, Latex; Section 8 has been modifie
Distribution of roots of random real generalized polynomials
The average density of zeros for monic generalized polynomials,
, with real holomorphic and
real Gaussian coefficients is expressed in terms of correlation functions of
the values of the polynomial and its derivative. We obtain compact expressions
for both the regular component (generated by the complex roots) and the
singular one (real roots) of the average density of roots. The density of the
regular component goes to zero in the vicinity of the real axis like
. We present the low and high disorder asymptotic
behaviors. Then we particularize to the large limit of the average density
of complex roots of monic algebraic polynomials of the form with real independent, identically distributed
Gaussian coefficients having zero mean and dispersion . The average density tends to a simple, {\em universal}
function of and in the domain where nearly all the roots are located for
large .Comment: 17 pages, Revtex. To appear in J. Stat. Phys. Uuencoded gz-compresed
tarfile (.66MB) containing 8 Postscript figures is available by e-mail from
[email protected]
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Factorization method and the supersymmetric monopole harmonics
We use the general
N
=1
supersymmetric formulation of one dimensional sigma models on nontrivial manifolds and its subsequent quantization to formulate the classical and quantum dynamics of the
N
=2
supersymmetric charged particle moving on a sphere in the field of a monopole. The factorization method is accommodated with the general covariance and it is used to integrate the corresponding system