Coherent Orthogonal Polynomials

We discuss as a fundamental characteristic of orthogonal polynomials like the existence of a Lie algebra behind them, can be added to their other relevant aspects. At the basis of the complete framework for orthogonal polynomials we put thus --in addition to differential equations, recurrence relations, Hilbert spaces and square integrable functions-- Lie algebra theory. We start here from the square integrable functions on the open connected subset of the real line whose bases are related to orthogonal polynomials. All these one-dimensional continuous spaces allow, besides the standard uncountable basis ${|x>}$, for an alternative countable basis ${|n>}$. The matrix elements that relate these two bases are essentially the orthogonal polynomials: Hermite polynomials for the line and Laguerre and Legendre polynomials for the half-line and the line interval, respectively. Differential recurrence relations of orthogonal polynomials allow us to realize that they determine a unitary representation of a non-compact Lie algebra, whose second order Casimir ${\cal C}$ gives rise to the second order differential equation that defines the corresponding family of orthogonal polynomials. Thus, the Weyl-Heisenberg algebra $h(1)$ with ${\cal C}=0$ for Hermite polynomials and $su(1,1)$ with ${\cal C}=-1/4$ for Laguerre and Legendre polynomials are obtained. Starting from the orthogonal polynomials the Lie algebra is extended both to the whole space of the ${\cal L}^2$ functions and to the corresponding Universal Enveloping Algebra and transformation group. Generalized coherent states from each vector in the space ${\cal L}^2$ and, in particular, generalized coherent polynomials are thus obtained.Comment: 11 page

Evaluation of the effect of training using auditory stimulation on rhythmic movement in Parkinsonian patients—a combined motor and [18F]-FDG PET study

[Abstract] Introduction: A programme of rehabilitation using auditory cues has previously been shown to decrease movement variability in the gait of Parkinsonian patients. Objective and methods: We studied the temporal variability of finger-tapping and gait in 9 patients with Parkinson's disease (PD) before and after they undertook a physical rehabilitation programme. Positron Emission Tomography (PET) using 2-deoxy-2[18F]fluoro-d-glucose (FDG) was performed in these subjects to look for changes in metabolic brain activity after completion of the rehabilitation program. Results: The reduction of variability was seen not only in gait but also other repetitive movements such as finger tapping. Furthermore, here we show differences in resting regional cerebral glucose utilisation in these patients compared to healthy controls (significant hypometabolism—p<0.001—for the PD group in the right parietal and temporal lobes, left temporal and frontal lobes and a hypermetabolism in the left cerebellum) and specific changes following the improvements in repetitive movement abilities (significant metabolic increment—p<0.001—in the PD group in the right cerebellum and in the right parietal and temporal lobes). Conclusions: Although our study does not allow us to draw firm conclusions, it provides new information on the neural basis of auditory stimulation in PD. Our results extend those from previous studies to show improvement in the temporal variability of two types of rhythmic movements after participation by PD patients in a physical rehabilitation programme, along with changes in glucose uptake in several brain areas involved in sensorimotor processing.Xunta de Galicia; PGIDIT02BTF13701P

Intertwining Symmetry Algebras of Quantum Superintegrable Systems

We present an algebraic study of a kind of quantum systems belonging to a family of superintegrable Hamiltonian systems in terms of shape-invariant intertwinig operators, that span pairs of Lie algebras like (su(n),so(2n)) or (su(p,q),so(2p,2q)). The eigenstates of the associated Hamiltonian hierarchies belong to unitary representations of these algebras. It is shown that these intertwining operators, related with separable coordinates for the system, are very useful to determine eigenvalues and eigenfunctions of the Hamiltonians in the hierarchy. An study of the corresponding superintegrable classical systems is also included for the sake of completness

Quantization on a 2-dimensional phase space with a constant curvature tensor

Some properties of the star product of the Weyl type (i.e. associated with the Weyl ordering) are proved. Fedosov construction of the *-product on a 2-dimensional phase spacewith a constant curvature tensor is presented. Eigenvalue equations for momentum p and position q on a 2-dimensional phase space with constant curvature tensors are solved.Comment: 33 pages, LaTeX, Annals of Physics (2003

A geometric approach to time evolution operators of Lie quantum systems

Lie systems in Quantum Mechanics are studied from a geometric point of view. In particular, we develop methods to obtain time evolution operators of time-dependent Schrodinger equations of Lie type and we show how these methods explain certain ad hoc methods used in previous papers in order to obtain exact solutions. Finally, several instances of time-dependent quadratic Hamiltonian are solved.Comment: Accepted for publication in the International Journal of Theoretical Physic

Classical and Quantum Integrable Systems in \wt{\gr{gl}}(2)^{+*} and Separation of Variables

Classical integrable Hamiltonian systems generated by elements of the Poisson commuting ring of spectral invariants on rational coadjoint orbits of the loop algebra \wt{\gr{gl}}^{+*}(2,{\bf R}) are integrated by separation of variables in the Hamilton-Jacobi equation in hyperellipsoidal coordinates. The canonically quantized systems are then shown to also be completely integrable and separable within the same coordinates. Pairs of second class constraints defining reduced phase spaces are implemented in the quantized systems by choosing one constraint as an invariant, and interpreting the other as determining a quotient (i.e., by treating one as a first class constraint and the other as a gauge condition). Completely integrable, separable systems on spheres and ellipsoids result, but those on ellipsoids require a further modification of order \OO(\hbar^2) in the commuting invariants in order to assure self-adjointness and to recover the Laplacian for the case of free motion. For each case - in the ambient space ${\bf R}^{n}$, the sphere and the ellipsoid - the Schr\"odinger equations are completely separated in hyperellipsoidal coordinates, giving equations of generalized Lam\'e type.Comment: 28 page

Geometrical origin of the *-product in the Fedosov formalism

The construction of the *-product proposed by Fedosov is implemented in terms of the theory of fibre bundles. The geometrical origin of the Weyl algebra and the Weyl bundle is shown. Several properties of the product in the Weyl algebra are proved. Symplectic and abelian connections in the Weyl algebra bundle are introduced. Relations between them and the symplectic connection on a phase space M are established. Elements of differential symplectic geometry are included. Examples of the Fedosov formalism in quantum mechanics are given.Comment: LaTeX, 39 page

Additional value of screening for minor genes and copy number variants in hypertrophic cardiomyopathy

Introduction: Hypertrophic cardiomyopathy (HCM) is the most prevalent inherited heart disease. Next-generation sequencing (NGS) is the preferred genetic test, but the diagnostic value of screening for minor and candidate genes, and the role of copy number variants (CNVs) deserves further evaluation. Methods: Three hundred and eighty-seven consecutive unrelated patients with HCM were screened for genetic variants in the 5 most frequent genes (MYBPC3, MYH7, TNNT2, TNNI3 and TPM1) using Sanger sequencing (N = 84) or NGS (N = 303). In the NGS cohort we analyzed 20 additional minor or candidate genes, and applied a proprietary bioinformatics algorithm for detecting CNVs. Additionally, the rate and classification of TTN variants in HCM were compared with 427 patients without structural heart disease. Results: The percentage of patients with pathogenic/likely pathogenic (P/LP) variants in the main genes was 33.3%, without significant differences between the Sanger sequencing and NGS cohorts. The screening for 20 additional genes revealed LP variants in ACTC1, MYL2, MYL3, TNNC1, GLA and PRKAG2 in 12 patients. This approach resulted in more inconclusive tests (36.0% vs. 9.6%, p<0.001), mostly due to variants of unknown significance (VUS) in TTN. The detection rate of rare variants in TTN was not significantly different to that found in the group of patients without structural heart disease. In the NGS cohort, 4 patients (1.3%) had pathogenic CNVs: 2 deletions in MYBPC3 and 2 deletions involving the complete coding region of PLN. Conclusions: A small percentage of HCM cases without point mutations in the 5 main genes are explained by P/LP variants in minor or candidate genes and CNVs. Screening for variants in TTN in HCM patients drastically increases the number of inconclusive tests, and shows a rate of VUS that is similar to patients without structural heart disease, suggesting that this gene should not be analyzed for clinical purposes in HCM