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There exist many attempts to define a Wigner function for qudits, each of them coming with its advantages and limitations. The existing finite versions have simple definitions, but they are artificial in their construction and do not allow an intuitive state analysis. The continuous versions have more complicated definitions, but they are similar to the original Wigner function and allow a visualization of the quantum states. The version based on the concept of tight frame we present is finite, but it has certain properties and applications similar to those of continuous versions. Based on the frame representation, we present several graphical representations of qubit states, and define two new parameters concerning them. We show that, from a mathematical point of view, the qubit is the orthogonal projection of qutrit., Comment: A new section called "Error detection and correction" has been added at page 18

Finite frame quantization is a discrete version of the coherent state quantization. In the case of a quantum system with finite-dimensional Hilbert space, the finite frame quantization allows us to associate a linear operator to each function defined on the discrete phase space of the system. We investigate the properties of the density operators which can be defined by using this method., Comment: 14 pages

The quantum systems with finite-dimensional Hilbert space have several applications and are intensively explored theoretically and experimentally. The mathematical description of these systems follows the analogy with the usual infinite-dimensional case. There exist finite versions for most of the elements used in the continuous case, but (to our knowledge) there does not exist a finite version corresponding to the mixed Gaussian states. Our aim is to fill this gap. The definition we propose for the mixed discrete Gaussian states is based on the explicit formulas we have obtained in the case of pure discrete variable Gaussian states., Comment: 12 pages, 1 figure. Any comments or suggestion are welcome. More details are available at https://unibuc.ro/user/nicolae.cotfas/

A remarkable discrete counterpart of the Gaussian function of one continuous variable can be defined by using a Jacobi theta function, that is, as the sum of a convergent series. We extend this approach to Gaussian functions of two variables, and investigate the Fourier transform and Wigner function of the functions of discrete variable defined in this way., Comment: 9 pages, 1 figure. For more details see the section "Documente" at https://unibuc.ro/user/nicolae.cotfas/

We show that a new unitary transform with characteristics almost similar to those of the finite Fourier transform can be defined in any finite-dimensional Hilbert space. It is defined by using the Kravchuk polynomials, and we call it Kravchuk transform. Some of its properties are investigated and used in order to obtain a simple alternative description for the irreducible representations of the Lie algebra su(2) and group SU(2). Our approach offers a deeper insight into the structure of the linear representations of SU(2) and new possibilities of computation, very useful in applications in quantum mechanics, quantum information, signal and image processing., Comment: Correction: reference to relation (33) replaced with the reference to relation (32) at the end of the proofs of Theorems 3 and 4

The mathematical description of the quantum harmonic oscillator is essentially based on the Gaussian function. In the case of a quantum oscillator with finite-dimensional Hilbert space, the position space consists in a finite number of points, and the ground state can be regarded as a finite version of the Gaussian function. Conversely, each finite Gaussian can be used in order to define some finite oscillators. We investigate certain finite Gaussians as concern their Fourier transform, Wigner function and associated oscillators., Comment: 23 pages, 6 figures

We analyze a generalized version of the Black-Scholes equation depending on a parameter $a\!\in \!(-\infty,0)$. It satisfies the martingale condition and coincides with the Black-Scholes equation in the limit case $a\nearrow 0$. We show that the generalized equation is exactly solvable in terms of Hermite polynomials and numerically compare its solution with the solution of the Black-Scholes equation.

The Black-Scholes model anticipates rather well the observed prices for options in the case of a strike price that is not too far from the current price of the underlying asset. Some useful extensions can be obtained by an adequate modification of the coefficients in the Black-Scholes equation. We investigate from a mathematical point of view an extension directly related to the quantum harmonic oscillator. In the considered case, the solution is the sum of a series involving the Hermite-Gauss functions. A finite-dimensional version is obtained by using a finite oscillator and the Harper functions. This simplified model keeps the essential characteristics of the continuous one and uses finite sums instead of series and integrals.

The Hamiltonian of the harmonic oscillator is usually defined as a differential operator, but an integral representation can be obtained by using the coherent state quantization. The finite frame quantization is a finite counterpart of the coherent state quantization and it allows us to define a finite oscillator by starting from the integral representation of the harmonic oscillator. Our purpose is to investigate the oscillator obtained in this way, and to present a possible application to the discrete fractional Fourier transform., Comment: Several new results and figures have been added

Weyl's formulation of quantum mechanics opened the possibility of studying the dynamics of quantum systems both in infinite-dimensional and finite-dimensional systems. Based on Weyl's approach, generalized by Schwinger, a self-consistent theoretical framework describing physical systems characterised by a finite-dimensional space of states has been created. The used mathematical formalism is further developed by adding finite-dimensional versions of some notions and results from the continuous case. Discrete versions of the continuous Gaussian functions have been defined by using the Jacobi theta functions. We continue the investigation of the properties of these finite Gaussians by following the analogy with the continuous case. We study the uncertainty relation of finite Gaussian states, the form of the associated Wigner quasi-distribution and the evolution under free-particle and quantum harmonic oscillator Hamiltonians. In all cases, a particular emphasis is put on the recovery of the known continuous-limit results when the dimension $d$ of the system increases., Comment: 21 pages, 4 figures

In this note we present an algorithm to generate new Schr\" odinger type equations explicitly solvable in terms of orthogonal polynomials or associated special functions., Comment: An example has been added

The generalization of the factorization method performed by Mielnik [J. Math. Phys. {\bf 25}, 3387 (1984)] opened new ways to generate exactly solvable potentials in quantum mechanics. We present an application of Mielnik's 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., Comment: New examples concerning the Poschl-Teller, Morse, Scarf, Coulomb, Rosen-Morse and Eckart type potentials have been added

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 the present 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 states quantizations are investigated. We also explore the localization aspects in the real line yielded by similar quantizations based on real Hermite polynomials., Comment: 15 pages, 6 figures

The hypergeometric type operators are shape invariant, and a factorization into a product of first order differential operators can be explicitly described in the general case. Some additional shape invariant operators depending on several parameters are defined in a natural way by starting from this general factorization. The mathematical properties of the eigenfunctions and eigenvalues of the operators thus obtained depend on the values of the involved parameters. We study the parameter dependence of orthogonality, square integrability and of the monotony of eigenvalue sequence. The obtained results allow us to define certain systems of Gazeau-Klauder coherent states and to describe some of their properties. Our systematic study recovers a number of well-known results in a natural unified way and also leads to new findings., Comment: An error occurring in Theorem 12 and Theorem 13 has been corrected

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., Comment: 28 pages, LaTeX in IOP style, New results added

The Penrose tiling is directly related to the atomic structure of certain decagonal quasicrystals and, despite its aperiodicity, is highly symmetric. It is known that the numbers 1, $-\tau $, $(-\tau)^2$, $(-\tau)^3$, ..., where $\tau =(1+\sqrt{5})/2$, are scaling factors of the Penrose tiling. We show that the set of scaling factors is much larger, and for most of them the number of the corresponding inflation centers is infinite., Comment: Paper submitted to Phil. Mag. (for Proceedings of Quasicrystals: The Silver Jubilee, Tel Aviv, 14-19 October, 2007)

Atomic-resolution electron microscope images show that a quasicrystal is a quasiperiodic packing of clusters. The outer atomic shells of multi-shell clusters occuring in quasicrystals are highly symmetric and rather robust, but some structural disorder may be present around the core. The models describing quasicrystal structure are quasiperiodic lattices containing two or more unit cells decorated by atomic clusters. We show that a modified version of the strip projection method may become an alternate way to obtain useful models., Comment: Paper for Proceedings of Aperiodic'06. Software and original of simulated diffraction patterns available at http://fpcm5.fizica.unibuc.ro/~ncotfas/

The diffraction image of a quasicrystal admits a finite group G as a symmetry group, and the quasicrystal can be regarded as a quasiperiodic packing of copies of a G-cluster C, joined by glue atoms. The physical space E containing C can be embedded into a higher-dimensional space R^k such that, up to an inflation factor, C is the orthogonal projection of the set {(\pm 1,0,...,0), (0,\pm 1,0,...,0), ... (0,...,0,\pm 1)}. The projections of the points of Z^k lying in the strip S=E+[-1/2,1/2]^k+t obtained by shifting a hypercube [-1/2,1/2]^k+t along E is a quasiperiodic packing of partially occupied copies of C, but unfortunately, the occupation of clusters is very low. In our modified strip projection method we firstly determine for each point x\in Z^k\cap S the number n(x) of all the arithmetic neighbours of x lying in the strip S, and project the points of Z^k\cap S on E in the decreasing order of the occupation number n(x). In the case when n(x) represents more than p% of all the points of the cluster C we project all the arithmetic neighbours of x (lying inside or outside S). We choose p such that to avoid the superposition of the fully occupied clusters. The projection of a point x with n(x) less than p% of all the points of the cluster C is added to the pattern only if it is not too close to the already obtained points., Comment: New figures concerning the Fourier transform available at http://fpcm5.fizica.unibuc.ro/~ncotfas/

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 considered shape invariant operators are directly related to Schrodinger type equations., Comment: More applications available at http://fpcm5.fizica.unibuc.ro/~ncotfas

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 considered equations are directly related to some Schrodinger type equations (Poschl-Teller, Scarf, Morse, etc), and the defined special functions are related to the corresponding bound-state eigenfunctions., Comment: Additional results available at http://fpcm5.fizica.unibuc.ro/~ncotfas

The positions of atoms forming a carbon nanotube are usually described by using a system of generators of the symmetry group. Each atomic position corresponds to an element of the set Z x {0,1,...,n} x {0,1}, where n depends on the considered nanotube. We obtain an alternate rather different description by starting from a three-axes description of the honeycomb lattice. In our mathematical model, which is a factor space defined by an equivalence relation in the set {(v_0,v_1,v_2)\in Z^3 | v_0+v_1+v_2\in {0,1}}, the neighbours of an atomic position can be described in a simpler way, and the mathematical objects with geometric or physical significance have a simpler and more symmetric form. We present some results concerning the linear representations of single-wall carbon nanotubes in order to illustrate the proposed approach., Comment: Major change of content. More details will be available at http://fpcm5.fizica.unibuc.ro/~ncotfas

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 some systems of coherent states. This general formalism allows us to extend some results known only in particular cases., Comment: Error corrected (section 5.2, page 7). More details are available at http://fpcm5.fizica.unibuc.ro/~ncotfas

Some of the most remarkable tilings and discrete quasiperiodic sets used in quasicrystal physics can be obtained by using strip projection method in a superspace of dimension four, five or six, and the projection of a unit hypercube as a window of selection. We present some mathematical results which allow one to use this very elegant method in superspaces of dimension much higher, and to generate discrete quasiperiodic sets with a more complicated local structure by starting from the corresponding covering cluster. Hundreds of points of these sets can be obtained in only a few minutes by using our computer programs., Comment: Paper accepted for publication in Philosophical Magazine. Software for quasicrystals available at http://fpcm5.fizica.unibuc.ro/~ncotfas/

Many of the mathematical models used in quasicrystal physics are based on tilings of the plane or space obtained by using strip projection method in a superspace of dimension four, five or six. We present some mathematical results which allow one to use this very elegant method in spaces of dimension much higher and to generate directly quasiperiodic packings of multi-shell clusters. We show that in the case of a two-dimensional (resp. three-dimensional) cluster we have to compute only determinants of order three (resp. four), independently of the dimension of the superspace we use. The computer program based on our mathematical results is very efficient. For example, we can easily generate quasiperiodic packings of three-shell icosahedral clusters (icosahedron + dodecahedron + icosidodecahedron) by using strip projection method in a 31-dimensional space (hundreds of points are obtained in a few minutes on a personal computer)., Comment: Computer program in FORTRAN 90 based on the strip projection method available on the website http://fpcm5.fizica.unibuc.ro/~ncotfas

The strip projection method is the most important way to generate quasiperiodic patterns with predefined local structure. We have obtained a very efficient algorithm for this method which allows one to use it in superspaces of very high dimension. A version of this algorithm for two-dimensional clusters and an application to decagonal two-shell clusters (strip projection in a 10-dimensional superspace) has been presented in math-ph/0504036. The program in FORTRAN 90 used in this case is very fast (700-800 points are obtained in 3 minutes). We present an application of our algorithm to three-dimensional clusters. The physical three-dimensional space is embedded into a 31-dimensional superspace and the strip projection method is used in order to generate a quasiperiodic packing of interpenetrating translated copies of a three-shell icosahedral cluster formed by the 12 vertices of a regular icosahedron (the first shell), the 20 vertices of a regular dodecahedron (the second shell) and the 30 vertices of an icosidodecahedron (the third shell). On a personal computer Pentium 4 with Fortran PowerStation version 4.0 (Microsoft Developer Studio) we obtain 400-500 points in 10 minutes. More details, bibliography and samples can be found on the website: http://fpcm5.fizica.unibuc.ro/~ncotfas/, Comment: Computer program in FORTRAN 90 and MATHEMATICA included

We present some mathematical results concerning the strip projection method and a computer program for generating quasiperiodic packings of decagonal two shell-clusters., Comment: Computer program in FORTRAN + MATHEMATICA included

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. This general formalism allows us to extend some known results to a larger class of functions., Comment: 10 pages

Model sets play a fundamental role in structure analysis of quasicrystals. The diffraction diagram of a quasicrystal admits as symmetry group a finite group G, and there is a G-cluster C (union of orbits of G) such that the quasicrystal can be regarded as a quasiperiodic packing of interpenetrating copies of C. We present an algorithm which leads from any G-cluster C directly to a multi-component model set Q such that the arithmetic neighbours of any point x belonging to Q are distributed on the sites of the translated copy x+C of C. Our mathematical algorithm may be useful in quasicrystal physics., Comment: This is the revised version (19 pages, 3 figures) of a paper submitted to Journal of Geometry and Physics. Software for generating quasiperiodic discrete sets available at http://fpcm5.fizica.unibuc.ro/~ncotfas

We show that the well known two-dimensional Penrose tiling admits an infinite number of independent scaling factors and an infinite number of inflation centers., Comment: 5 pages, LaTeX2e in IOP Style

The positions of atoms forming a carbon nanotube are usually described by using a system of generators of the symmetry group. Each atomic position corresponds to an element of the set Z X {0,1,...,n} X {0,1}, where n is a natural number depending on the considered nanotube. We obtain an alternate rather different description by starting from a description of the honeycomb lattice in terms of Miller indices. In our mathematical model which is a factor space defined by an equivalence relation in the set {(v_0,v_1,v_2)\in Z^3 | v_0+v_1+v_2\in {0,1}} the neighbours of an atomic position can be described in a simpler way, and the mathematical objects with geometric or physical significance have a simpler and more symmetric form., Comment: 14 pages, 2 figures, LaTeX2e

A quasiperiodic packing Q of interpenetrating copies of C, most of them only partially occupied, can be defined in terms of the strip projection method for any icosahedral cluster C. We show that in the case when the coordinates of the vectors of C belong to the quadratic field Q[\sqrt{5}] the dimension of the superspace can be reduced, namely, Q can be re-defined as a multi-component model set by using a 6-dimensional superspace., Comment: 7 pages, LaTeX2e in IOP style

The diffraction pattern of a quasicrystal admits as symmetry group a finite group G, and there exists a G-cluster C (a union of orbits of G) such that the quasicrystal can be regarded as a quasiperiodic packing of copies of C, generally, partially occupied. On the other hand, by starting from the G-cluster C we can define in a canonical way a permutation representation of G in a higher dimensional space, decompose this space into the orthogonal sum of two G-invariant subspaces and use the strip projection method in order to define a pattern which can also be regarded as a quasiperiodic packing of copies of C, generally, partially occupied. This mathematical algorithm is useful in quasicrystal physics, but the dimension of the superspace we have to use in the case of a two or three-shell cluster is rather large. We show that the generalization concerning the strip projection method proposed by Baake and Moody [Proc. Int. Conf. Aperiodic' 97 (Alpe d'Huez, 27-31 August, 1997) ed M de Boissieu et al. (Singapore: World Scientific, 1999) pp 9-20] allows to reduce this dimension, and present some examples., Comment: 11 pages, LaTeX2e in IOPP style

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 considered equations are directly related to some Schrodinger type equations (Poschl-Teller, Scarf, Morse, etc), and the defined special functions are related to the corresponding bound-state eigenfunctions., Comment: 10 pages, LaTeX2e in IOP style

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., Comment: 4 pages, presented at XXIV International Colloquium on Group Theoretical Methods in Physics (Paris, July 15-20, 2002)