L-classical d-orthogonal polynomial sets of Sheffer type

In this paper, we characterize L-classical d-orthogonal polynomial sets of Sheffer type where L being a lowering operator commutating with the derivative operator D and belonging to {D,eD-1, sin(D)}. For the first case we state a (d+1)-order differential equation satisfied by the corresponding polynomials. We, also, show that, with these three lowering operators, all the orthogonal polynomial sets are classified as L-classical orthogonal polynomial sets.

MENENTUKAN PERSAMAAN GELOMBANG SCHR�DINGER POTENSIAL NON-SENTRAL SCARF HIPERBOLIK PLUS ROSEN-MORSE TRIGONOMETRIK MENGGUNAKANMETODE SUPERSIMETRI MEKANIKA KUANTUM

Penelitian ini bertujuan untuk menentukan persamaan gelombang Schrdinger potensial non-sentral Scarf hiperbolik plus Rosen-Morse trigonometrik menggunakan metode Supersimetri Mekanika Kuantum (SUSI MK). Persamaangelombang radial diperoleh dari persamaan Schrdinger bagian radial, sedangkan persamaan gelombang sudut diperoleh dari persamaan Schrdinger bagian sudut polar. Penentuan persamaan gelombang tingkat dasar ditentukan dengan sifat lowering operator dan persamaan gelombang tereksitasi ditentukan dengan sifat raising operator. Jadi, baik untuk bagian radial maupun bagian polar ditentukan dengan menggunakan metode operator supersimetri. Adapun tampilan gambar dari fungsi gelombang bagian polar menggunakan aplikasi program maple 12.

Arik–Coon q-oscillator cat states on the noncommutative complex plane ℂq−1 and their nonclassical properties

The normalized even and odd [Formula: see text]-cat states corresponding to Arik–Coon [Formula: see text]-oscillator on the noncommutative complex plane [Formula: see text] are constructed as the eigenstates of the lowering operator of a [Formula: see text]-deformed [Formula: see text] algebra with the left eigenvalues. We present the appropriate noncommutative measures in order to realize the resolution of the identity condition by the even and odd [Formula: see text]-cat states. Then, we obtain the [Formula: see text]-Bargmann–Fock realizations of the Fock representation of the [Formula: see text]-deformed [Formula: see text] algebra as well as the inner products of standard states in the [Formula: see text]-Bargmann representations of the even and odd subspaces. Also, the Euler’s formula of the [Formula: see text]-factorial and the Gaussian integrals based on the noncommutative [Formula: see text]-integration are obtained. Violation of the uncertainty relation, photon antibunching effect and sub-Poissonian photon statistics by the even and odd [Formula: see text]-cat states are considered in the cases [Formula: see text] and [Formula: see text].

Solution of The Schrödinger Equation for Trigonometric Scarf Plus Poschl-Teller Non-Central Potential Using Supersymmetry Quantum Mechanics

<span>In this paper, we show that the exact energy eigenvalues and eigen functions of the Schrödinger <span>equation for charged particles moving in certain class of noncentral potentials can be easily <span>calculated analytically in a simple and elegant manner by using Supersymmetric method <span>(SUSYQM). We discuss the trigonometric Scarf plus Poschl-Teller systems. Then, by operating <span>the lowering operator we get the ground state wave function, and the excited state wave functions <span>are obtained by operating raising operator repeatedly. The energy eigenvalue is expressed in the <span>closed form obtained using the shape invariant properties. The results are in exact agreement with <span>other methods.</span></span></span></span></span></span></span><br /></span>

RANK LOWERING OPERATORS ON GL(n, ℝ)

If one takes the Mellin transform of an automorphic form for GL(n) and then integrates it along the diagonal on GL(n - 1) then one obtains an automorphic form on GL(n - 1). This gives a rank lowering operator. In this paper a more general rank lowering operator is obtained by combining the Mellin transform with a sum of powers of certain fixed differential operators. The analytic continuation of the rank lowering operator is obtained by showing that the spectral expansion consists of sums of Rankin–Selberg L-functions of type GL(n) × GL(n - 1).

NEW DISCRETE STATES IN TWO-DIMENSIONAL SUPERGRAVITY

Two-dimensional string theory is known to contain the set of discrete states that are the SU (2) multiplets generated by the lowering operator of the SU (2) current algebra. Their structure constants are defined by the area preserving diffeomorphisms in two dimensions. In this paper we show that the interaction of d = 2 superstrings with the superconformal β - γ ghosts enlarges the actual algebra of the dimension 1 currents and hence the new ghost-dependent discrete states appear. Generally, these states are the SU (N) multiplets if the algebra includes the currents of ghost numbers n : -N ≤ n ≤ N - 2, not related by picture changing. We compute the structure constants of these ghost-dependent discrete states for N = 3 and express them in terms of SU (3) Clebsch–Gordan coefficients, relating this operator algebra to the volume preserving diffeomorphisms in d = 3. For general N, the operator algebra is conjectured to be isomorphic to SDiff (N). This points at possible holographic relations between two-dimensional superstrings and field theories in higher dimensions.