We derive some analytic closed-form solutions for a class of Riccati equations y'(x) - λ0(x)y(x) ± y2(x) = ±s0(x) where λ0(x), s0(x) are C∞-functions. We show that if δn = λnsn-1 - λn-1sn = 0, where λn = λ'n-1 + sn-1 + λ0λn-1and sn = s'n-1 + s0λk-1, n = 1, 2, ..., then the Riccati equation has a solution given by y(x) = sn-1(x)/λn-1(x). Extension to the generalized Riccati equation y'(x) + P(x)y(x) + Q(x)y2(x) = R(x) is also investigated.

We develop a variational method to obtain accurate bounds for the eigenenergies of H = -Δ + V in arbitrary dimensions N > 1, where V(r) is the nonpolynomial oscillator potential . The variational bounds are compared with results previously obtained in the literature. An infinite set of exact solutions is also obtained and used as a source of comparison eigenvalues.

In this paper we derive expressions for matrix elements (&phgr;i, H&phgr;j) for the Hamiltonian H = −&Dgr; + ∑qa(q)rq in d ⩾ 2 dimensions. The basis functions in each angular momentum subspace are of the form . The matrix elements are given in terms of the Gamma function for all d. The significance of the parameters t and p and scale s are discussed. Applications to a variety of potentials are presented, including potentials with singular repulsive terms of the form &bgr;/r&agr;, &agr;, &bgr; > 0, perturbed Coulomb potentials −D/r + Br + Ar2, and potentials with weak repulsive terms, such as −&ggr;r2 + r4, &ggr; > 0.

Infinite series of the typeare investigated. Closed-form sums are obtained for α a positive integer, α &equal; 1, 2, 3, &ldots;. The limiting case of b → &infty;, after y is replaced with x2/b, leads toThis type of series appears in the first-order perturbation correction for the wavefunction of the generalized spiked harmonic oscillator HamiltonianThese results have immediate applications to perturbation series for the energy and wavefunction of the spiked harmonic oscillator Hamiltonian

We apply the asymptotic iteration method (AIM) (Ciftci, Hall and Saad 2003 J. Phys. A: Math. Gen. 36 11807) to solve new classes of second-order homogeneous linear differential equation. In particular, solutions are found for a general class of eigenvalue problems which includes Schrödinger problems with Coulomb, harmonic oscillator or Pöschl–Teller potentials, as well as the special eigenproblems studied recently by Bender et al (2001 J. Phys. A: Math. Gen. 34 9835) and generalized in the present paper to arbitrary dimension.

We consider the differential equations y″ = λ0(x)y' + s0(x)y, where λ0(x), s0(x) are C∞-functions. We prove (i) if the differential equation has a polynomial solution of degree n > 0, then δn = λnsn-1 - λn-1sn = 0, where λn = λ'n-1 + sn-1 + λ0λn-1andsn = s'n-1 + s0λk-1, n = 1, 2, .... Conversely (ii) if λnλn-1 ≠ 0 and δn = 0, then the differential equation has a polynomial solution of degree at most n. We show that the classical differential equations of Laguerre, Hermite, Legendre, Jacobi, Chebyshev (first and second kinds), Gegenbauer and the Hypergeometric type, etc obey this criterion. Further, we find the polynomial solutions for the generalized Hermite, Laguerre, Legendre and Chebyshev differential equations.

Infinite series∑n&equal;1&infty; [(α/2)n/n] [1/n!] 1F1(−n, γ, x2),where 1F1(−n, γ, x2) &equal; [n!/(γ)n] Ln(γ−1)(x2),appear in the first-order perturbation correction for the wavefunction of the generalized spiked harmonic oscillator Hamiltonian H &equal; −d2/dx2 + Bx2 + A/x2 + λ/xα, 0 ≤ x < &infty;, α, λ > 0, A ≥ 0. It is proved that the series is convergent for all x > 0 and 2γ > α where γ &equal; 1 + ?surd;(1 + 4A). Closed-form sums are presented for these series for the cases α &equal; 2, 4 and 6. A general formula for finding the sum for α/2 &equal; 2 + m, m &equal; 0, 1, 2, &ldots; in terms of associated Laguerre polynomials is also provided.

A variational analysis is presented for the generalized spiked harmonic oscillator Hamiltonian operator -d2/dx2+Bx2+A/ x2+/x, where is a real positive parameter. The formalism makes use of a basis provided by exact solutions of Schrödinger's equation for the Gol'dman and Krivchenkov Hamiltonian, and the corresponding matrix elements that were previously found. For all the discrete eigenvalues the method provides bounds which improve as the dimension D of the basis set is increased. Extension to the N-dimensional case in arbitrary angular momentum subspaces is also presented. By minimizing over the free parameter A, we are able to reduce substantially the number of basis functions needed for a given accuracy.

The eigenvalue bounds obtained earlier (Hall R L and Saad N 1998 J. Phys. A: Math. Gen. 31 963) for smooth transformations of the form are extended to N dimensions. In particular a simple formula is derived which bounds the eigenvalues for the spiked harmonic oscillator potential , , , and is valid for all discrete eigenvalues, arbitrary angular momentum l and spatial dimension N.

Under certain constraints on the parameters a, b and c, it is known that Schrödinger's equation -d2ψ/dx2 + (ax6 + bx4 + cx2)ψ = Eψ, a > 0, with the sextic anharmonic oscillator potential is exactly solvable. In this paper we show that the exact wavefunction ψ is the generating function for a set of orthogonal polynomials {P(t)n(x)} in the energy variable E. Some of the properties of these polynomials are discussed in detail and our analysis reveals scaling and factorization properties that are central to quasi-exact solvability. We also prove that this set of orthogonal polynomials can be reduced, by means of a simple scaling transformation, to a remarkable class of orthogonal polynomials, Pn(E) = P(0)n(E) recently discovered by Bender and Dunne.

We study weak-coupling perturbation expansions for theground-state energy of the Hamiltonian with the generalizedspiked harmonic oscillator potential V(x) = Bx2 + (A/x2) + (λ/xα), and also for the bottoms of theangular-momentum subspaces labelled by l = 0,1,... , inN dimensions corresponding to the spiked harmonic oscillatorpotential V(x) = x2 + (λ/xα), where α isa real positive parameter. A method of Znojil (Znojil M 1993J. Math. Phys. 34 4914) is then applied to obtain closed-formexpressions for the sums of some infinite series whose termsinvolve ratios and products of gamma functions.

A variational and perturbative treatment is provided for afamily of generalized spiked harmonic oscillator HamiltoniansH = -d2/dx2 + Bx2 + A/x2 + λ/xα, where B>0, A≥0, and α and λdenote two real positive parameters. The method makes use of thefunction space spanned by the solutions |n⟩ ofSchrödinger's equation for the potential V(x) = Bx2 + A/x2. Compact closed-form expressions are obtained for thematrix elements ⟨m|H|n⟩, and a first-order perturbationseries is derived for the wavefunction. The results are givenin terms of generalized hypergeometric functions. It is provedthat the series for the wavefunction is absolutely convergentfor α≤2.

Classes of solvable potentials are presented within an standard application of supersymmetric quantum mechanics. Sets of exceptional orthogonal polynomials generated by these solvable potentials are introduced and examined in detail. Several properties of these polynomials including orthogonality conditions, weight functions, differential equations, the Wronskains, possible recurrence relations are also investigated.

Infinite series of the type ∞ ∑ n=1 ( 2 )n n 1 n! 2F1(−n, b； γ； y) are investigated. Closed-form sums are obtained for α a positive integer α = 1, 2, 3, . . . . The limiting case of b → ∞, after y is replaced with x2/b, leads to ∞ ∑ n=1 ( 2 )n n 1 n! 1F1(−n, γ, x2). This type of series appears in the first-order perturbation correction for the wavefunction of the generalized spiked harmonic oscillator Hamiltonian H = − d dx ； Bx2 ； A x ； λ x 0 ≤ x < ∞, α, λ > 0, A ≥ 0. These results have immediate applications to perturbation series for the energy and wave function of the spiked harmonic oscillator Hamiltonian H = − d dx ；Bx2 ； λ x 0 ≤ x < ∞, α, λ > 0. PACS 03.65.Ge

Abstract A class of orthogonal polynomials in two quaternionic variables is introduced. This class serves as an analogous to the classical Zernike polynomials Z m , n ( β ) ( z , z ¯ ) ( arXiv:1502.07256 , 2014). A number of interesting properties such as the orthogonality condition, recurrence relations, raising and lowering operators are discussed in detail. Particularly, the ladder operators, realized as differential operators in terms of the so-called Cullen derivatives, for these quaternionic polynomials are also studied. Some physically interesting summation and integral formulas are also proved, and their physical relevance briefly discussed.