Nicolae Cotfas

Complex and real Hermite polynomials and related quantizations

It is known that the anti-Wick (or standard coherent state) quantization of the complex plane produces both canonical commutation rule and quantum spectrum of the harmonic oscillator (up to the addition of a constant). In this work, we show that these two issues are not necessarily coupled: there exists a family of separable Hilbert spaces, including the usual Fock–Bargmann space, and in each element in this family there exists an overcomplete set of unit-norm states resolving the unity. With the exception of the Fock–Bargmann case, they all produce non-canonical commutation relation whereas the quantum spectrum of the harmonic oscillator remains the same up to the addition of a constant. The statistical aspects of these non-equivalent coherent state quantizations are investigated. We also explore the localization aspects in the real line yielded by similar quantizations based on real Hermite polynomials.

Finite tight frames and some applications

A finite-dimensional Hilbert space is usually described in terms of an orthonormal basis, but in certain approaches or applications a description in terms of a finite overcomplete system of vectors, called a finite tight frame, may offer some advantages. The use of a finite tight frame may lead to a simpler description of the symmetry transformations, to a simpler and more symmetric form of invariants or to the possibility to define new mathematical objects with physical meaning, particularly in regard with the notion of a quantization of a finite set. We present some results concerning the use of integer coefficients and frame quantization, several examples and suggest some possible applications.

Finite-dimensional Hilbert space and frame quantization

The quantum observables used in the case of quantum systems with finite-dimensional Hilbert space are defined either algebraically in terms of an orthonormal basis and discrete Fourier transformation or by using a continuous system of coherent states. We present an alternative approach to these important quantum systems based on the finite frame quantization. Finite systems of coherent states, usually called finite tight frames, can be defined in a natural way in the case of finite quantum systems. Novel examples of such tight frames are presented. The quantum observables used in our approach are obtained by starting from certain classical observables described by functions defined on the discrete phase space corresponding to the system. They are obtained by using a finite frame and a Klauder–Berezin–Toeplitz-type quantization. Semi-classical aspects of tight frames are studied through lower symbols of basic classical observables.

Systems of orthogonal polynomials defined by hypergeometric type equations with application to quantum mechanics

A hypergeometric type equation satisfying certain conditions defines either a finite or an infinite system of orthogonal polynomials. We present in a unified and explicit way all these systems of orthogonal polynomials, the associated special functions and the corresponding raising/lowering operators. The equations considered are directly related to some Schrödinger type equations (Pöschl-Teller, Scarf, Morse, etc), and the special functions defined are related to the corresponding bound-state eigenfunctions.

Shape invariance, raising and lowering operators in hypergeometric type equations

The Schrodinger equations which are exactly solvable in terms of associated special functions are directly related to some self-adjoint operators defined in the theory of hypergeometric type equations. The fundamental formulae occurring in a supersymmetric approach to these Hamiltonians are consequences of some formulae concerning the general theory of associated special functions. We use this connection in order to obtain a general theory of Schrodinger equations exactly solvable in terms of associated special functions, and to extend certain results known in the case of some particular potentials.

Permutation Representations Defined by G-Clusters with Application to Quasicrystals

Starting from each finite union of orbits (called G- cluster) of an R-irreducible orthogonal representation of a finite group G, we define a representation of G in a higher-dimensional space (called permutation representation), and we prove that it can be decomposed into an orthogonal sum of two representations such that one of them is equivalent to the initial representation. This decomposition allows us to use the strip projection method and to obtain some patterns useful in quasicrystal physics. We show that certain self-similarities of such a pattern can be obtained by using the decomposition into R-irreducible components of the corresponding permutation representation, and we present two examples.

Shape-invariant hypergeometric type operators with application to quantum mechanics

A hypergeometric type equation satisfying certain conditions defines either a finite or an infinite system of orthogonal polynomials. The associated special functions are eigenfunctions of some shape-invariant operators. These operators can be analysed together and the mathematical formalism we use can be extended in order to define other shape-invariant operators. All the shape-invariant operators considered are directly related to Schrödinger-type equations.

Random walks on carbon nanotubes and quasicrystals

An analytic formula for the total number of k -step walks between given sites on a carbon nanotube is obtained by using a new mathematical model based on a three-axes description of the honeycomb lattice. The new model represents an alternate mathematical description which may be useful in certain applications. It is similar to the four-axes description existing in the case of hexagonal crystals. The use of one or more additional axes is a fundamental method in quasicrystal physics. We show that the mathematical model we use for the honeycomb lattice can be defined in terms of a strip projection method, and present a method to associate some finite graphs to a quasicrystal. The random walks on these graphs are connected with random walks on a quasicrystal.

G-model sets and their self-similarities

The model sets (also called cut and project sets), first defined by Yves Meyer in harmonic analysis, play a central role in quasicrystal modelling. Each of them is defined by using a cut and project scheme containing two projectors and a lattice. We present a method which can be used to study the self-similarities of a model set based on the matrices of these projectors in a basis of the lattice. This method also allows one to study the self-similarities of the diffraction spectrum of a model set because, generally, the Bragg peaks with intensity above a given threshold also form a model set. The diffraction pattern corresponding to a quasicrystal is invariant under a finite group G, and the local structure of the quasicrystal can be described by using a finite union of orbits of G, called a G cluster. The neighbours of each atom belong to some orbits of G, and the quasicrystal can be regarded as a union of interpenetrating partially occupied translations of the corresponding G cluster. We present a method to obtain a model set (called the G-model set) by starting from a G cluster. The experimental diffraction patterns allow one to determine the symmetry group G, and high-resolution electron microscopy images enable one to choose a G cluster describing the local structure. The existing computer programs for the cut and project method allow one to pass directly from the local structure of the quasicrystal to a mathematical model, to compute the theoretical diffraction spectrum and to compare it with the experimental data.

Hypergeometric type operators and their supersymmetric partners

The generalization of the factorization method performed by Mielnik [J. Math. Phys. 25, 3387 (1984)] opened new ways to generate exactly solvable potentials in quantum mechanics. We present an application of Mielniks method to hypergeometric type operators. It is based on some solvable Riccati equations and leads to a unitary description of the quantum systems exactly solvable in terms of orthogonal polynomials or associated special functions.