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Richard L. Hall

Publications and source records attributed to Richard L. Hall.

At least 55 records · Page 3Linked to original sources
arXiv2008

The Klein-Gordon equation with the Kratzer potential in d dimensions

We apply the Asymptotic Iteration Method to obtain the bound-state energy spectrum for the d-dimensional Klein-Gordon equation with scalar S(r) and vector potentials V(r). When S(r) and V(r) are both Coulombic, we obtain all the exact solutions; when the potentials are both of Kratzer type, we obtain all the exact solutions for S(r)=V(r); if S(r) > V(r) we obtain exact solutions under certain constraints on the potential parameters: in this case, a possible general solution is found in terms of a monic polynomial, whose coefficients form a set of elementary symmetric polynomials.

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arXiv2007

Eigenvalue bounds for polynomial central potentials in d dimensions

If a single particle obeys non-relativistic QM in R^d and has the Hamiltonian H = - Delta + f(r), where f(r)=sum_{i = 1}^{k}a_ir^{q_i}, 2\leq q_i < q_{i+1}, a_i \geq 0$, then the eigenvalues E = E_{n\ell}^{(d)}(λ) are given approximately by the semi-classical expression E = \min_{r > 0}[\frac{1}{r^2} + \sum_{i = 1}^{k}a_i(P_ir)^{q_i}]. It is proved that this formula yields a lower bound if P_i = P_{n\ell}^{(d)}(q_1), an upper bound if $P_i = P_{n\ell}^{(d)}(q_k) and a general approximation formula if P_i = P_{n\ell}^{(d)}(q_i). For the quantum anharmonic oscillator f(r)=r^2+λr^{2m},m=2,3,... in d dimension, for example, E = E_{n\ell}^{(d)}(λ) is determined by the algebraic expression λ={1\over β}({2α(m-1)\over mE-δ})^m({4α\over (mE-δ)}-{E\over (m-1)}) where δ={\sqrt{E^2m^2-4α(m^2-1)}} and α, βare constants. An improved lower bound to the lowest eigenvalue in each angular-momentum subspace is also provided. A comparison with the recent results of Bhattacharya et al (Phys. Lett. A, 244 (1998) 9) and Dasgupta et al (J. Phys. A: Math. Theor., 40 (2007) 773) is discussed.

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arXiv2007

Solutions to the 1d Klein-Gordon equation with cutoff Coulomb potentials

In a recent paper by Barton (J. Phys. A40, 1011 (2007)), the 1-dimensional Klein-Gordon equation was solved analytically for the non-singular Coulomb-like potential V_1(|x|) = -α/(|x|+a). In the present paper, these results are completely confirmed by a numerical formulation that also allows a solution for an alternative cutoff Coulomb potential V_2(|x|) = -α/|x|, ~|x| > a, and otherwise V_2(|x|) = -α/a.

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arXiv2007

Ultrarelativistic N-boson systems

General analytic energy bounds are derived for N-boson systems governed by ultrarelativistic Hamiltonians of the form H = sum_{i=1}^N||p_i|| + sum_{1=i<j}^N V(r_{ij}), where V(r) is a static attractive pair potential. It is proved that a translation-invariant model Hamiltonian H_c provides a lower bound to H for all N \ge 2. This result was conjectured in an earlier paper but proved only for N = 2,3,4. As an example, the energy in the case of the linear potential V(r) = r is determined with error less than 0.55% for all N \ge 2.

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arXiv2007

Binding energies of semirelativistic N-boson systems

General analytic energy bounds are derived for N-boson systems governed by semirelativistic Hamiltonians of the form H=\sum_{i=1}^N \sqrt(p_i^2+m^2) + \sum_{1=i \ge generally, and we prove this for N=3, and for N=4 when m=0. The conjecture is also valid generally for the harmonic oscillator and in the nonrelativistic large-m limit. This formulation allows reductions to scaled 3- or 4-body problems, whose spectral bottoms provide energy lower bounds. The example of the ultrarelativistic linear potential is studied in detail and explicit upper- and lower-bound formulas are derived and compared with earlier bounds.

math-ph↗
arXiv2006

Criterion for polynomial solutions to a class of linear differential equation of second order

We consider the differential equations y''=λ_0(x)y'+s_0(x)y, where λ_0(x), s_0(x) are C^{\infty}-functions. We prove (i) if the differential equation, has a polynomial solution of degree n >0, then δ_n=λ_n s_{n-1}-λ_{n-1}s_n=0, where λ_{n}= λ_{n-1}^\prime+s_{n-1}+λ_0λ_{n-1}\hbox{and}\quad s_{n}=s_{n-1}^\prime+s_0λ_{k-1},\quad n=1,2,.... Conversely (ii) if λ_nλ_{n-1}\ne 0 and δ_n=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 kind), 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.

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arXiv2006

Gravitating semirelativistic N-boson systems

Analytic energy bounds for N-boson systems governed by semirelativistic Hamiltonians of the form H=\sum_{i=1}^N(p_i^2 + m^2)^{1/2} - sum_{1=i 0, are derived by use of Jacobi relative coordinates. For gravity v=c/N, these bounds are substantially tighter than earlier bounds and they are shown to coincide with known results in the nonrelativistic limit.

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arXiv2006

Study of a class of non-polynomial oscillator potentials

We develop a variational method to obtain accurate bounds for the eigenenergies of H = -Delta + V in arbitrary dimensions N>1, where V(r) is the nonpolynomial oscillator potential V(r) = r^2 + lambda r^2/(1+gr^2), lambda in (-infinity,\infinity), g>0. 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.

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arXiv2006

Sextic anharmonic oscillators and orthogonal polynomials

Under certain constraints on the parameters a, b and c, it is known that Schroedinger's equation -y"(x)+(ax^6+bx^4+cx^2)y(x) = E y(x), a > 0, with the sextic anharmonic oscillator potential is exactly solvable. In this article we show that the exact wave function y is the generating function for a set of orthogonal polynomials P_n^{(t)}(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, P_n(E)=P_n^{(0)}(E) recently discovered by Bender and Dunne.

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arXiv2005

Exact solutions for semirelativistic problems with non-local potentials

It is shown that exact solutions may be found for the energy eigenvalue problem generated by the class of semirelativistic Hamiltonians of the form H = sqrt{m^2+p^2} + hat{V}, where hat{V} is a non-local potential with a separable kernel of the form V(r,r') = - sum_{i=1}^n v_i f_i(r)g_i(r'). Explicit examples in one and three dimensions are discussed, including the Yamaguchi and Gauss potentials. The results are used to obtain lower bounds for the energy of the corresponding N-boson problem, with upper bounds provided by the use of a Gaussian trial function.

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arXiv2005

Schroedinger upper bounds to semirelativistic eigenvalues

Problems posed by semirelativistic Hamiltonians of the form H = sqrt{m^2+p^2} + V(r) are studied. It is shown that energy upper bounds can be constructed in terms of certain related Schroedinger operators; these bounds include free parameters which can be chosen optimally.

math-ph↗
arXiv2005

Green's function for a Schroedinger operator and some related summation formulas

Summation formulas are obtained for products of associated Lagurre polynomials by means of the Green's function K for the Hamiltonian H = -{d^2\over dx^2} + x^2 + Ax^{-2}, A > 0. K is constructed by an application of a Mercer type theorem that arises in connection with integral equations. The new approach introduced in this paper may be useful for the construction of wider classes of generating function.

math-ph↗
arXiv2005

Iterative solutions to the Dirac equation

We consider a single particle which is bound by a central potential and obeys the Dirac equation in d dimensions. We first apply the asymptotic iteration method to recover the known exact solutions for the pure Coulomb case. For a screened-Coulomb potential and for a Coulomb plus linear potential with linear scalar confinement, the method is used to obtain accurate approximate solutions for both eigenvalues and wave functions.

math-ph↗
arXiv2005

Perturbation theory in a framework of iteration methods

In a previous paper (J. Phys. A 36, 11807 (2003)), we introduced the `asymptotic iteration method' for solving second-order homogeneous linear differential equations. In this paper, 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.

math-ph↗
arXiv2004

Construction of exact solutions to eigenvalue problems by the asymptotic iteration method

We apply the asymptotic iteration method (AIM) [J. Phys. A: Math. Gen. 36, 11807 (2003)] 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 Schroedinger problems with Coulomb, harmonic oscillator, or Poeschl-Teller potentials, as well as the special eigenproblems studied recently by Bender et al [J. Phys. A: Math. Gen. 34 9835 (2001)] and generalized in the present paper to higher dimensions.

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arXiv2004

Study of anharmonic singular potentials

A simple and efficient variational method is introduced to accelerate the convergence of the eigenenergy computations for a Hamiltonian H with singular potentials. Closed-form analytic expressions in N dimensions are obtained for the matrix elements of H with respect to the eigenfunctions of a soluble singular problem with two free parameters A and B. The matrix eigenvalues are then optimized with respect to A and B for a given N. Applications, convergence rates, and comparisons with earlier work are discussed in detail.

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