SUSY transformations with complex factorization constants. Application to spectral singularities

Supersymmetric (SUSY) transformation operators corresponding to complex factorization constants are analyzed as operators acting in the Hilbert space of functions square integrable on the positive semiaxis. Obtained results are applied to Hamiltonians possessing spectral singularities which are non-Hermitian SUSY partners of selfadjoint operators. A new regularization procedure for the resolution of the identity operator in terms of continuous biorthonormal set of the non-Hermitian Hamiltonian eigenfunctions is proposed. It is also shown that the continuous spectrum eigenfunction has zero binorm (in the sense of distributions) at the singular point.Comment: Thanks to A. Sokolov a number of inaccuracies are correcte

SUSY transformation of the Green function and a trace formula

An integral relation is established between the Green functions corresponding to two Hamiltonians which are supersymmetric (SUSY) partners and in general may possess both discrete and continuous spectra. It is shown that when the continuous spectrum is present the trace of the difference of the Green functions for SUSY partners is a finite quantity which may or may not be equal to zero despite the divergence of the traces of each Green function. Our findings are illustrated by using the free particle example considered both on the whole real line and on a half line

Binary Patterns in Binary Cube-Free Words: Avoidability and Growth

The avoidability of binary patterns by binary cube-free words is investigated and the exact bound between unavoidable and avoidable patterns is found. All avoidable patterns are shown to be D0L-avoidable. For avoidable patterns, the growth rates of the avoiding languages are studied. All such languages, except for the overlap-free language, are proved to have exponential growth. The exact growth rates of languages avoiding minimal avoidable patterns are approximated through computer-assisted upper bounds. Finally, a new example of a pattern-avoiding language of polynomial growth is given.Comment: 18 pages, 2 tables; submitted to RAIRO TIA (Special issue of Mons Days 2012

Electromagnetic form factors of the (rho) meson in light cone QCD sum rules

We investigate the electromagnetic form factors of the (rho) meson in light cone QCD sum rules. We find that the ratio of the magnetic and charge form factors is larger than two at all values of Q^2, (Q^2 >= 0.5 GeV^2). The values of the individual form factors at fixed values of Q^2 predicted by the light cone QCD sum rules are quite different compared to the results of other approaches. These results can be checked in future, when more precise data on (rho) meson form factors is available.Comment: 12 pages, 6 figures, LaTeX formatte

Initial-state nuclear effects in proton-nucleus collisions

Two important initial-state nuclear effects in hadron-nucleus collisions are considered. The ratios of inclusive differential cross sections for Drell-Yan dimuon production are calculated. The calculated results are compared to the E866 data. It is shown that consideration of multiple soft rescatterings of incident quarks in nuclei and initial-state quark energy loss effects allow to get a good agreement between the calculated results and the experimental data.Comment: 6 pages, 6 figure

Is the CPT-norm always positive?

We give an explicit example of an exactly solvable PT-symmetric Hamiltonian with the unbroken PT symmetry which has one eigenfunction with the zero PT-norm. The set of its eigenfunctions is not complete in corresponding Hilbert space and it is non-diagonalizable. In the case of a regular Sturm-Liouville problem any diagonalizable PT-symmetric Hamiltonian with the unbroken PT symmetry has a complete set of positive CPT-normalazable eigenfunctions. For non-diagonalizable Hamiltonians a complete set of CPT-normalazable functions is possible but the functions belonging to the root subspace corresponding to multiple zeros of the characteristic determinant are not eigenfunctions of the Hamiltonian anymore

On Abelian repetition threshold

We study the avoidance of Abelian powers of words and consider three reasonable generalizations of the notion of Abelian power to fractional powers. Our main goal is to find an Abelian analogue of the repetition threshold, i.e., a numerical value separating k-avoidable and k-unavoidable Abelian powers for each size k of the alphabet. We prove lower bounds for the Abelian repetition threshold for large alphabets and all definitions of Abelian fractional power. We develop a method estimating the exponential growth rate of Abelian-power-free languages. Using this method, we get non-trivial lower bounds for Abelian repetition threshold for small alphabets. We suggest that some of the obtained bounds are the exact values of Abelian repetition threshold. In addition, we provide upper bounds for the growth rates of some particular Abelian-power-free languages. © 2011 EDP Sciences

Eigenphase preserving two-channel SUSY transformations

We propose a new kind of supersymmetric (SUSY) transformation in the case of the two-channel scattering problem with equal thresholds, for partial waves of the same parity. This two-fold transformation is based on two imaginary factorization energies with opposite signs and with mutually conjugated factorization solutions. We call it an eigenphase preserving SUSY transformation as it relates two Hamiltonians, the scattering matrices of which have identical eigenphase shifts. In contrast to known phase-equivalent transformations, the mixing parameter is modified by the eigenphase preserving transformation.Comment: 16 pages, 1 figur

Exact propagators for SUSY partners

Pairs of SUSY partner Hamiltonians are studied which are interrelated by usual (linear) or polynomial supersymmetry. Assuming the model of one of the Hamiltonians as exactly solvable with known propagator, expressions for propagators of partner models are derived. The corresponding general results are applied to "a particle in a box", the Harmonic oscillator and a free particle (i.e. to transparent potentials).Comment: 31 page