Tamas Palmai

Interlacing of positive real zeros of Bessel functions

We unify the three distinct inequality sequences (Abramowitz and Stegun (1972) [1, 9.5.2]) of positive real zeros of Bessel functions into a single one.

On nonsingular potentials of Cox–Thompson inversion scheme

We establish a condition for obtaining nonsingular potentials using the Cox–Thompson inverse scattering method with one phase shift. The anomalous singularities of the potentials are avoided by maintaining unique solutions of the underlying Regge–Newton integral equation for the transformation kernel. As a by-product, new inequality sequences of zeros of Bessel functions are discovered.

Regularization of multi-soliton form factors in sine-Gordon model

Abstract A general and systematic regularization is developed for the exact solitonic form factors of exponential operators in the ( 1 + 1 )-dimensional sine-Gordon model by analytical continuation of their integral representations. The procedure is implemented in Mathematica. Test results are shown for four- and six-soliton form factors. Program summary Program title: SGFF Catalogue identifier: AEMG_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEMG_v1_0.html Program obtainable from: CPC Program Library, Queenʼs University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 1462 No. of bytes in distributed program, including test data, etc.: 15 488 Distribution format: tar.gz Programming language: Mathematica [1] Computer: PC Operating system: Cross-platform Classification: 7.7, 11.1, 23 Nature of problem: The multi-soliton form factors of the sine-Gordon model (relevant in two-dimensional physics) were given only by highly non-trivial integral representation with a limited domain of convergence. Practical applications of the form factors, e.g. calculation of correlation functions in two-dimensional condensed matter systems, were not possible in general. Solution method: Using analytic continuation techniques an efficient algorithm is found and implemented in Mathematica, which provides a general and systematic way to calculate multi-soliton form factors in the sine-Gordon model. The package contains routines to compute the two-, four- and six-soliton form factors. Running time: Strongly dependent on the desired accuracy and the number of solitons. For physical rapidities after an initialization of about 30 s, the calculation of the two-, four- and six-soliton form factors at a single point takes approximately 0.5 s, 2.5 s and 8 s, respectively. Reference: [1] Wolfram Research, Inc., Mathematica Edition: Version 7.0, Wolfram Research, Inc., Champaign, Illinois, 2008.

Renyi entanglement entropies of descendant states in critical systems with boundaries: conformal field theory and spin chains

We discuss the Renyi entanglement entropies of descendant states in critical one-dimensional systems with boundaries, that map to boundary conformal field theories in the scaling limit. We unify the previous conformal-field-theory approaches to describe primary and descendant states in systems with both open and closed boundaries. We provide universal expressions for the first two descendants in the identity family. We apply our technique to critical systems belonging to different universality classes with non-trivial boundary conditions that preserve conformal invariance, and find excellent agreement with numerical results obtained for finite spin chains. We also demonstrate that entanglement entropies are a powerful tool to resolve degeneracy of higher excited states in critical lattice models.

SEMI-ANALYTIC EQUATIONS TO THE COX THOMPSON INVERSE SCATTERING METHOD AT FIXED ENERGY FOR SPECIAL CASES

Solution of the Cox–Thompson inverse scattering problem at fixed energy1–3 is reformulated resulting in semi-analytic equations. The new set of equations for the normalization constants and the nonphysical (shifted) angular momenta are free of matrix inversion operations. This simplification is a result of treating only the input phase shifts of partial waves of a given parity. Therefore, the proposed method can be applied for identical particle scattering of the bosonic type (or for certain cases of identical fermionic scattering). The new formulae are expected to be numerically more efficient than the previous ones. Based on the semi-analytic equations an approximate method is proposed for the generic inverse scattering problem, when partial waves of arbitrary parity are considered.

Simplified solutions of the Cox–Thompson inverse scattering method at fixed energy

Simplified solutions of the Cox–Thompson inverse quantum scattering method at fixed energy are derived if a finite number of partial waves with only even or odd angular momenta contribute to the scattering process. Based on new formulae various approximate methods are introduced which also prove applicable to the generic scattering events.

Fixed energy potentials through an auxiliary inverse eigenvalue problem

An inverse scattering method based on an auxiliary inverse Sturm–Liouville problem recently proposed by Horvath and Apagyi (2008 Mod. Phys. Lett. B 22 2137) is examined in various aspects and developed further to (re)construct spherically symmetric fixed energy potentials of compact support realized in the three-dimensional Schrodinger equation. The method is generalized to obtain a family of inverse procedures characterized by two parameters originating, respectively, from the Liouville transformation and the solution of the inverse Sturm–Liouville problem. Both parameters affect the bound states arising in the auxiliary inverse spectral problem and one of them enables us to reduce their number which is assessed by a simple method. Various solution techniques of the underlying moment problem are proposed including the exact Cauchy matrix inversion method, usage of spurious bound state and assessment of the number of bound states. Examples include (re)productions of potentials from phase shifts known theoretically or derived from scattering experiments.

Development of a Cox?Thompson inverse scattering method to charged particles

A Cox–Thompson fixed-energy quantum inverse scattering method is developed further to treat long-range Coulomb interaction. Depending on the reference potentials chosen, two methods have been formulated which produce inverse potentials with singular or finite value at the origin. Based on the quality of reproduction of input experimental phase shifts, it is guessed that the p–α interaction possesses an interesting repulsive hard core.

Study of the Cox–Thompson inverse scattering method with a Coulomb potential

In order to learn more about the precision of the inversion by the Cox?Thompson method, we investigated the inversion of phase shifts of a singular potential, namely of a Coulomb potential. Using asymptotically Riccati?Bessel functions as reference functions, we could only approximately reproduce the singularity of the Coulomb potential at the origin. We also show uncertainties in the inverted potential due to different minima in the minimization solution of the nonlinear equations of the Cox?Thompson procedure. As a result, we conclude that one has to take much care with the inversion of experimental phase shifts suffering from measurement errors.