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The exact solutions of Schrödinger's equation with the deformed Hulthén potential Vq(x) = −μ e−δ x/(1 − q e−δ x), δ, μ, q > 0 are given, along with a closed-form formula for the normalization constants of the eigenfunctions for arbitrary q > 0. The Crum-Darboux transformation is then used to derive the corresponding exact solutions for the extended Hulthén potentials V (x) = − μ e − δ x / (1 − q e − δ x ) + q j (j + 1) e − δ x / (1 − q e − δ x ) 2 , j = 0,1,2 , .... A general formula for the new normalization condition is also provided. [ABSTRACT FROM AUTHOR]

An analysis of the generalized confluent Heun equation (α2r2 + α1r) y″ + (β2r2 + β1r + β0) y′ − (ε1r + ε0) y = 0 in d-dimensional space, where {αi, βi, εi} are real parameters, is presented. With the aid of these general results, the quasi-exact solvability of the Schrödinger eigenproblem generated by the softcore Coulomb potential V(r) = −e2Z/(r + b), b > 0, is explicitly resolved. Necessary and sufficient conditions for polynomial solvability are given. A three-term recurrence relation is provided to generate the coefficients of polynomial solutions explicitly. We prove that these polynomial solutions are sources of finite sequences of orthogonal polynomials. Properties such as the recurrence relations, Christoffel-Darboux formulas, and moments of the weight function are discussed. We also reveal a factorization property of these polynomials which permits the construction of other interesting related sequences of orthogonal polynomials. [ABSTRACT FROM AUTHOR]

The function E = F(v) expresses the dependence of a discrete eigenvalue E of the Schrödinger Hamiltonian H = -Δ + vf(r) on the coupling parameter v. We use envelope theory to generate a functional sequence {f [k](r)} to reconstruct f(r) from F(v) starting from a seed potential f [0](r). In the power-law or log cases, the inversion can be effected analytically and is complete in just two steps. In other cases, convergence is observed numerically. To provide concrete illustrations of the inversion method it is first applied to the Hulthén potential, and it is then used to invert spectral data generated by singular potentials with shapes of the form f(r) = -a/r + b sgn(q)rq and f(r) = -a/r + bln (r), a, b > 0. For the class of attractive central potentials with shapes f(r) = g(r)/r, with g(0) < 0 and g′(r) >= 0, we prove that the ground-state energy curve F(v) determines f(r) uniquely. [ABSTRACT FROM AUTHOR]

Exact solutions to the d-dimensional Schrödinger equation, d >= 2, for Coulomb plus harmonic oscillator potentials V(r) = -a/r + br2, b > 0, and a ≠ 0 are obtained. The potential V(r) is considered both in all space, and under the condition of spherical confinement inside an impenetrable spherical box of radius R. With the aid of the asymptotic iteration method, the exact analytic solutions under certain constraints, and general approximate solutions, are obtained. These exhibit the parametric dependence of the eigenenergies on a, b, and R. The wave functions have the simple form of a product of a power function, an exponential function, and a polynomial. Polynomial solutions are found for differential equations of the form py″ + qy′ - ry = 0, where p, q, and r are given polynomials with degrees 4, 3, and 2, respectively. [ABSTRACT FROM AUTHOR]

The asymptotic iteration method (AIM) is applied to obtain highly accurate eigenvalues of the radial Schrödinger equation with the singular potential V(r) = r2+λ/rα(α,λ>0) in arbitrary dimensions. Certain fundamental conditions for the application of AIM, such as a suitable asymptotic form for the wave function, and the termination condition for the iteration process, are discussed. Several suggestions are introduced to improve the rate of convergence and to stabilize the computation. AIM offers a simple, accurate, and efficient method for the treatment of singular potentials, such as V(r), valid for all ranges of coupling λ. [ABSTRACT FROM AUTHOR]

Abstract: In a previous paper [J. Phys. A 36 (2003) 11807], we introduced the ‘asymptotic iteration method’ for solving second-order homogeneous linear differential equations. In this Letter, we study perturbed problems in quantum mechanics and we use the method to find the coefficients in the perturbation series for the eigenvalues and eigenfunctions directly, without first solving the unperturbed problem. [Copyright &y& Elsevier]

Examines the lowest energy E of a relativistic systems of N identitical bosons bounded by harmonic-oscillator pair potentials in three spatial dimensions. Derivation of energy bounds; Limits of the lower and upper bound for all N identical bosons; Application of relative coordinates on the energy bounds in the Schrödinger limit.

The method of potential envelopes is used to analyse the bound-state spectrum of the Schro¨dinger Hamiltonian H=−Δ−v/(r+b), where v and b are positive. We established simple formulas yielding upper and lower energy bounds for all the energy eigenvalues. [Copyright &y& Elsevier]

Focuses on the inversion inequality for potentials in quantum mechanics. Supposition that the ground-state eigenvalue E=F(v) of the Schrodinger Hamiltonian in one dimension is known for all values of the coupling v>0; Expression of the potential shape in the form f(x)=G(x[sup 2]); Establishment of the inversion inequality.

It is shown that the discrete spectrum of Schrödinger Hamiltonians of the form H=-Δ+vf may be represented by the semiclassical expression Enl=minr>=0 {K(f)nl(r) + vf(r)}. The K functions are found to be invariant with respect to coupling and shifts: K(Af+B)=K(f). For pure power laws, f(r)=sgn(q)rq, and the log potential, they are also invariant with respect to scale, and have the simple forms (Pnl(q)/r)2 and (Lnl/r)2, respectively. K functions are also derived for sech-squared and Hulthén potentials. If f=g(h), where g is a smooth transformation, then the envelope approximation is expressed in terms of K by the relation K(f)[bar_over_tilde:_approx._equal_to]K(h). When the transformation g has definite convexity, then the approximation immediately yields eigenvalue bounds for all n and l. The theory is used to prove the log-power theorem Lnl = Pnl(0), which, in turn, generates a simple eigenvalue formula for the log potential. [ABSTRACT FROM AUTHOR]