Discrete Darboux transformation for discrete polynomials of hypergeometric type

Darboux Transformation, well known in second order differential operator theory, is applied here to the difference equation satisfied by the discrete hypergeometric polynomials(Charlier, Meixner-Krawchuk, Hahn)

Quasi-exactly solvable problems and the dual (q-)Hahn polynomials

A second-order differential (q-difference) eigenvalue equation is constructed whose solutions are generating functions of the dual (q-)Hahn polynomials. The fact is noticed that these generating functions are reduced to the (little q-)Jacobi polynomials, and implications of this for quasi-exactly solvable problems are studied. A connection with the Azbel-Hofstadter problem is indicated.Comment: 15 pages, LaTex; final version, presentation improved, title changed, to appear in J.Math.Phy

q-Ultraspherical polynomials for q a root of unity

Properties of the $q$-ultraspherical polynomials for $q$ being a primitive root of unity are derived using a formalism of the $so_q(3)$ algebra. The orthogonality condition for these polynomials provides a new class of trigonometric identities representing discrete finite-dimensional analogs of $q$-beta integrals of Ramanujan.Comment: 7 pages, LATE

Advanced Method for Forecasting and Warning of Severe Convective Weather and Local-scale Hazards

Hurricane Ida ferociously affected many south-eastern and eastern parts of the United States, making it one of the strongest hurricanes in recent years. Advanced forecast and warning tool has been used to track the path of the ex-Hurricane, Ida, as it left New Orleans on its way towards the northeast, accurately predicting significant supercell development above New York City on September 01, 2021. This advanced method accurately detected the area with the highest possible level of convective instability with 24-h lead time and even Level 5, devised in the categorical outlooks legend of the system. Therefore, an extreme level implied a very high probability of the local-scale hazard occurring above the NYC. Cloud model output fields (updrafts and downdrafts, wind shear, near-surface convergence, the vertical component of relative vorticity) show the rapid development of a strong supercell storm with rotating updrafts and a mesocyclone. The characteristic hook-shaped echo signature visible in the reflectivity patterns indicates a signal for a highly precipitable (HP) supercell with the possibility of tornado initiation. Open boundary conditions represent a good basis for simulating a tornado that evolved from a supercell storm, initialized with initial data obtained from a real-time simulation in the period when the bow echo and tornado-like signature occurred. Тhe modeled results agree well with the observations

Self-Similar Potentials and the q-Oscillator Algebra at Roots of Unity

Properties of the simplest class of self-similar potentials are analyzed. Wave functions of the corresponding Schr\"odinger equation provide bases of representations of the $q$-deformed Heisenberg-Weyl algebra. When the parameter $q$ is a root of unity the functional form of the potentials can be found explicitly. The general $q^3=1$ and the particular $q^4=1$ potentials are given by the equianharmonic and (pseudo)lemniscatic Weierstrass functions respectively.Comment: 15 pp, Latex, to appear in Lett.Math.Phy

An SU(2) Analog of the Azbel--Hofstadter Hamiltonian

Motivated by recent findings due to Wiegmann and Zabrodin, Faddeev and Kashaev concerning the appearence of the quantum U_q(sl(2)) symmetry in the problem of a Bloch electron on a two-dimensional magnetic lattice, we introduce a modification of the tight binding Azbel--Hofstadter Hamiltonian that is a specific spin-S Euler top and can be considered as its ``classical'' analog. The eigenvalue problem for the proposed model, in the coherent state representation, is described by the S-gap Lam\'e equation and, thus, is completely solvable. We observe a striking similarity between the shapes of the spectra of the two models for various values of the spin S.Comment: 19 pages, LaTeX, 4 PostScript figures. Relation between Cartan and Cartesian deformation of SU(2) and numerical results added. Final version as will appear in J. Phys. A: Math. Ge

Localization of N=4 Superconformal Field Theory on S^1 x S^3 and Index

We provide the geometrical meaning of the ${\cal N}=4$ superconformal index. With this interpretation, the ${\cal N}=4$ superconformal index can be realized as the partition function on a Scherk-Schwarz deformed background. We apply the localization method in TQFT to compute the deformed partition function since the deformed action can be written as a $\delta_\epsilon$-exact form. The critical points of the deformed action turn out to be the space of flat connections which are, in fact, zero modes of the gauge field. The one-loop evaluation over the space of flat connections reduces to the matrix integral by which the ${\cal N}=4$ superconformal index is expressed.Comment: 42+1 pages, 2 figures, JHEP style: v1.2.3 minor corrections, v4 major revision, conclusions essentially unchanged, v5 published versio

Integrable (2+1)-dimensional systems of hydrodynamic type

We describe the results that have so far been obtained in the classification problem for integrable (2+1)-dimensional systems of hydrodynamic type. The systems of Gibbons--Tsarev type are the most fundamental here. A whole class of integrable (2+1)-dimensional models is related to each such system. We present the known GT systems related to algebraic curves of genus g=0 and g=1 and also a new GT system corresponding to algebraic curves of genus g=2. We construct a wide class of integrable models generated by the simplest GT system, which was not considered previously because it is in a sense trivial.Comment: 47 pages, no figure

Darboux transformations of coherent states of the time-dependent singular oscillator

Darboux transformation of both Barut-Girardello and Perelomov coherent states for the time-dependent singular oscillator is studied. In both cases the measure that realizes the resolution of the identity operator in terms of coherent states is found and corresponding holomorphic representation is constructed. For the particular case of a free particle moving with a fixed value of the angular momentum equal to two it is shown that Barut-Giriardello coherent states are more localized at the initial time moment while the Perelomov coherent states are more stable with respect to time evolution. It is also illustrated that Darboux transformation may keep unchanged this different time behavior.Comment: 13 page

Charging the Superconformal Index

The superconformal index is an important invariant of superconformal field theories. In this note we refine the superconformal index by inserting the charge conjugation operator C. We construct a matrix integral for this charged index for N=4 SYM with SU(N) gauge group. The key ingredient for the construction is a "charged character," which reduces to Tr(C) for singlet representations of the gauge group. For each irreducible real SU(N) representation, we conjecture that this charged character is equal to the standard character for a corresponding representation of SO(N+1) or SP(N-1), for N even or odd respectively. The matrix integral for the charged index passes tests for small N and for N -> infinity. Like the ordinary superconformal index, for N=4 SYM the charged index is independent of N in the large-N limit.Comment: 31 pages, v2: minor changes, published versio