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F. Cannata

Publications and source records attributed to F. Cannata.

At least 19 recordsLinked to original sources
arXiv2015

New Implicitly Solvable Potential Produced by Second Order Shape Invariance

The procedure proposed recently by J.Bougie, A.Gangopadhyaya and J.V.Mallow to study the general form of shape invariant potentials in one-dimensional Supersymmetric Quantum Mechanics (SUSY QM) is generalized to the case of Higher Order SUSY QM with supercharges of second order in momentum. A new shape invariant potential is constructed by this method. It is singular at the origin, it grows at infinity, and its spectrum depends on the choice of connection conditions in the singular point. The corresponding Schrödinger equation is solved explicitly: the wave functions are constructed analytically, and the energy spectrum is defined implicitly via the transcendental equation which involves Confluent Hypergeometric functions.

quant-ph↗
arXiv2012

Three-Dimensional Shape Invariant Non-Separable Model With Equidistant Spectrum

A class of three-dimensional models which satisfy supersymmetric intertwining relations with the simplest - oscillator-like - variant of shape invariance is constructed. It is proved that the models are not amenable to conventional separation of variables for the complex potentials, but their spectra are real and equidistant (like for isotropic harmonic oscillator). The special case of such potential with quadratic interaction is solved completely. The Hamiltonian of the system is non-diagonalizable, and its wave functions and associated functions are built analytically. The symmetry properties of the model and degeneracy of energy levels are studied.

hep-th↗
arXiv2012

Equidistance of the Complex 2-Dim Anharmonic Oscillator Spectrum: Exact Solution

We study a class of quantum two-dimensional models with complex potentials of specific form. They can be considered as the generalization of a recently studied model with quadratic interaction not amenable to conventional separation of variables. In the present case, the property of shape invariance provides the equidistant form of the spectrum and the algorithm to construct eigenfunctions analytically. It is shown that the Hamiltonian is non-diagonalizable, and the resolution of identity must include also the corresponding associated functions. In the specific case of anharmonic second-plus-fourth order interaction, expressions for the wave functions and associated functions are constructed explicitly for the lowest levels, and the recursive algorithm to produce higher level wave functions is given.

math-ph↗
arXiv2011

New Two-Dimensional Quantum Models with Shape Invariance

Two-dimensional quantum models which obey the property of shape invariance are built in the framework of polynomial two-dimensional SUSY Quantum Mechanics. They are obtained using the expressions for known one-dimensional shape invariant potentials. The constructed Hamiltonians are integrable with symmetry operators of fourth order in momenta, and they are not amenable to the conventional separation of variables.

hep-th↗
arXiv2010

Spectral singularities for Non-Hermitian one-dimensional Hamiltonians: puzzles with resolution of identity

We examine the completeness of bi-orthogonal sets of eigenfunctions for non-Hermitian Hamiltonians possessing a spectral singularity. The correct resolutions of identity are constructed for delta like and smooth potentials. Their form and the contribution of a spectral singularity depend on the class of functions employed for physical states. With this specification there is no obstruction to completeness originating from a spectral singularity.

math-ph↗
arXiv2010

Exactly Solvable Non-Separable and Non-Diagonalizable 2-Dim Model with Quadratic Complex Interaction

We study a quantum model with non-isotropic two-dimensional oscillator potential but with additional quadratic interaction $x_1x_2$ with imaginary coupling constant. It is shown, that for a specific connection between coupling constant and oscillator frequences, the model {\it is not} amenable to a conventional separation of variables. The property of shape invariance allows to find analytically all eigenfunctions and the spectrum is found to be equidistant. It is shown that the Hamiltonian is non-diagonalizable, and the resolution of the identity must include also the corresponding associated functions. These functions are constructed explicitly, and their properties are investigated. The problem of $R-$separation of variables in two-dimensional systems is discussed.

hep-th↗
arXiv2009

Phantom without phantom or how the PT symmetry saves us from the Big Rip

We consider the PT symmetric flat Friedmann model of two scalar fields with positive kinetic terms. While the potential of one ("normal") field is taken real, that of the other field is complex. We study a complex classical solution of the system of the two Klein-Gordon equations together with the Friedmann equation. The solution for the normal field is real while the solution for the second field is purely imaginary, realizing classically the "phantom" behavior. The energy density and pressure are real and the corresponding geometry is well-defined. The Lagrangian for the linear perturbations has the correct potential signs for both the fields, so that the problem of stability does not arise. The background dynamics is determined by an effective action including two real fields one normal and one "phantom". Remarkably, the phantom phase in the cosmological evolution is transient and the Big Rip never occurs. Our model is contrasted to well-known quintom models, which also include one normal and one phantom fields.

gr-qc↗
arXiv2009

Three-Dimensional Solutions of Supersymmetrical Intertwining Relations and Pairs of Isospectral Hamiltonians

The general solution of SUSY intertwining relations for three-dimensional Schrödinger operators is built using the class of second order supercharges with nondegenerate constant metric. This solution includes several models with arbitrary parameters. We are interested only in quantum systems which are not amenable to separation of variables, i.e. can not be reduced to lower dimensional problems. All constructed Hamiltonians are partially integrable - each of them commutes with a symmetry operator of fourth order in momenta. The same models can be considered also for complex values of parameters leading to a class of non-Hermitian isospectral Hamiltonians.

hep-th↗
arXiv2008

Two-field cosmological models and large-scale cosmic magnetic fields

We consider two different toy cosmological models based on two fields (one normal scalar and one phantom) realizing the same evolution of the Bang-to-Rip type. One of the fields (pseudoscalar) interacts with the magnetic field breaking the conformal invariance of the latter. The effects of the amplification of cosmic magnetic fields are studied and it is shown that the presence of such effects can discriminate between different cosmological models realizing the same global evolution of the universe.

hep-th↗
arXiv2008

Signals of bimodality in the fragmentation of Au quasi-projectiles

Signals of bimodality have been investigated in experimental data of quasi-projectile decay produced in Au+Au collisions at 35 AMeV. This same data set was already shown to provide several signals characteristic of a first order, liquid-gas-like phase transition. Different event sortings proposed in the recent literature are analyzed. A sudden change in the fragmentation pattern is revealed by the distribution of the charge of the largest fragment, compatible with a bimodal behavior.

nucl-ex↗
arXiv2007

Two-dimensional Schrödinger Hamiltonians with Effective Mass in SUSY Approach

The general solution of SUSY intertwining relations of first order for two-dimensional Schrödinger operators with position-dependent (effective) mass is built in terms of four arbitrary functions. The procedure of separation of variables for the constructed potentials is demonstrated in general form. The generalization for intertwining of second order is also considered. The general solution for a particular form of intertwining operator is found, its properties - symmetry, irreducibility, separation of variables - are investigated.

hep-th↗
arXiv2007

Symmetries and the cosmological constant puzzle

We outline the evaluation of the cosmological constant in the framework of the standard field-theoretical treatment of vacuum energy and discuss the relation between the vacuum energy problem and the gauge-group spontaneous symmetry breaking. We suggest possible extensions of the 't Hooft-Nobbenhuis symmetry, in particular, its complexification till duality symmetry and discuss the compatible implementation on gravity. We propose to use the discrete time-reflection transform to formulate a framework in which one can eliminate the huge contributions of vacuum energy into the effective cosmological constant and suggest that the breaking of time--reflection symmetry could be responsible for a small observable value of this constant.

gr-qc↗
arXiv2007

Pseudo-Hermiticity of an Exactly Solvable Two-Dimensional Model

We study a two-dimensional exactly solvable non-Hermitian $PT-$non-symmetric quantum model with real spectrum, which is not amenable to separation of variables, by supersymmetrical methods. Here we focus attention on the property of pseudo-Hermiticity, biorthogonal expansion and pseudo-metric operator. To our knowledge this is the first time that pseudo-Hermiticity is realized explicitly for a nontrivial two-dimensional case. It is shown that the Hamiltonian of the model is not diagonalizable.

hep-th↗
arXiv2007

Non-linear Supersymmetry for non-Hermitian, non-diagonalizable Hamiltonians: I. General properties

We study complex potentials and related non-diagonalizable Hamiltonians with special emphasis on formal definitions of associated functions and Jordan cells. The nonlinear SUSY for complex potentials is considered and the theorems characterizing its structure are presented. We present the class of potentials invariant under SUSY transformations for non-diagonalizable Hamiltonians and formulate several results concerning the properties of associated functions . We comment on the applicability of these results for softly non-Hermitian PT-symmetric Hamiltonians. The role of SUSY (Darboux) transformations in increasing/decreasing of Jordancells in SUSY partner Hamiltonians is thoroughly analyzed and summarized in the Index Theorem. The properties of non-diagonalizable Hamiltonians as well as the Index Theorem are illustrated in the solvable examples of non-Hermitian reflectionless Hamiltonians . The rigorous proofs are relegated to the Part II of this paper. At last, some peculiarities in resolution of identity for discrete and continuous spectra with a zero-energy bound state at threshold are discussed.

math-ph↗
arXiv2006

Bimodal pattern in the fragmentation of Au quasi-projectiles

Signals of bimodality have been investigated in experimental data of quasi-projectile decay produced in Au+Au collisions at 35 AMeV. This same data set was already shown to present several signals characteristic of a first order, liquid-gas-like phase transition. For the present analysis, events are sorted in bins of transverse energy of light charged particles emitted by the quasi-target source. A sudden change in the fragmentation pattern is observed from the distributions of the asymmetry of the two largest fragments, and the charge of the largest fragment. This latter distribution shows a bimodal behavior. The interpretation of this signal is discussed.

nucl-ex↗