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Richard L. Hall

Publications and source records attributed to Richard L. Hall.

At least 73 records · Page 4Linked to original sources
arXiv2004

A basis for variational calculations in d dimensions

In this paper we derive expressions for matrix elements (ϕ_i,Hϕ_j) for the Hamiltonian H=-Δ+\sum_q a(q)r^q in d > 1 dimensions. The basis functions in each angular momentum subspace are of the form phi_i(r)=r^{i+1+(t-d)/2}e^{-r^p/2}, i >= 0, p > 0, t > 0. 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 b/r^a, a,b > 0, perturbed Coulomb potentials -D/r + B r + Ar^2, and potentials with weak repulsive terms, such as -g r^2 + r^4, g > 0.

math-ph↗
arXiv2004

Relativistic N-boson systems bound by pair potentials V(r_{ij}) = g(r_{ij}^2)

We study the lowest energy E of a relativistic system of N identical bosons bound by pair potentials of the form V(r_{ij}) = g(r_{ij}^2) in three spatial dimensions. In natural units hbar = c = 1 the system has the semirelativistic `spinless-Salpeter' Hamiltonian H = \sum_{i=1}^N \sqrt{m^2 + p_i^2} + \sum_{j>i=1}^N g(|r_i - r_j|^2), where g is monotone increasing and has convexity g'' >= 0. We use `envelope theory' to derive formulas for general lower energy bounds and we use a variational method to find complementary upper bounds valid for all N >= 2. In particular, we determine the energy of the N-body oscillator g(r^2) = c r^2 with error less than 0.15% for all m >= 0, N >= 2, and c > 0.

math-ph↗
arXiv2004

Phase shifts and resonances in the Dirac equation

We review the analytic results for the phase shifts delta_{l}(k) in non-relativistic scattering from a spherical well. The conditions for the existence of resonances are established in terms of time-delays. Resonances are shown to exist for p-waves (and higher angular momenta) but not for s-waves. These resonances occur when the potential is not quite strong enough to support a bound p-wave of zero energy. We then examine relativistic scattering by spherical wells and barriers in the Dirac equation. In contrast to the non-relativistic situation, s-waves are now seen to possess resonances in scattering from both wells and barriers. When s-wave resonances occur for scattering from a well, the potential is not quite strong enough to support a zero momentum s-wave solution at E = m. Resonances resulting from scattering from a barrier can be explained in terms of the `crossing' theorem linking s-wave scattering from barriers to p-wave scattering from wells. A numerical procedure to extract phase shifts for general short range potentials is introduced and illustrated by considering relativistic scattering from a Gaussian potential well and barrier.

math-ph↗
arXiv2003

Friedrichs extensions of Schroedinger operators with singular potentials

The Friedrichs extension for the generalized spiked harmonic oscillator given by the singular differential operator -D^2+ Bx^2 + Ax^{-2} + lambda x^{-alpha} (B>0, A >= 0) in L_2(0, infinity) is studied. We look at two different domains of definition for each of these differential operators in L_2(0, infinity), namely C_0^infinity(0, infinity) and D(T_{2,F})\cap D(M_{lambda, alpha}), where the latter is a subspace of the Sobolev space W_{2,2}(0, infinity). Adjoints of these differential operators on C_0^infinity(0,infinity) exist as result of the null-space properties of functionals. For the other domain, convolutions and Jensen and Minkowski integral inequalities, density of C_0^\infinity(0,\infinity) in D(T_{2,F})\cap D(M_{λ, α}) in L_2(0,\infinity) lead to the other adjoints. Further density properties C_0^infinity(0,infinity) on D(T_{2,F})\cap D(M_{λ, α}) yield the Friedrichs extension of these differential operators with domains of definition D(T_{2,F})\cap D(M_{lambda, alpha}).

math-ph↗
arXiv2003

The energy of a system of relativistic massless bosons bound by oscillator pair potentials

We study the lowest energy E of a semirelativistic system of N identical massless bosons with Hamiltonian H= sum{i=1 to N} sqrt(p_i^2)+ sum{j>i=1 to N} g |r_i - r_j|^2, g > 0. We prove the inequalities A [g N^2 (N-1)^2]^{1/3} < E < B [g N^2 (N-1)^2]^{1/3}, where A = 2.33810741 and B = [81/(2 pi)]^{1/3} = 2.3447779. The average of these bounds determines E with an error less than 0.15% for all N > 1.

math-ph↗
arXiv2003

Perturbation expansions for a class of singular potentials

Harrell's modified perturbation theory [Ann. Phys. 105, 379-406 (1977)] is applied and extended to obtain non-power perturbation expansions for a class of singular Hamiltonians H = -D^2 + x^2 + A/x^2 + lambda/x^alpha, (A\geq 0, alpha > 2), known as generalized spiked harmonic oscillators. The perturbation expansions developed here are valid for small values of the coupling lambda > 0, and they extend the results which Harrell obtained for the spiked harmonic oscillator A = 0. Formulas for the the excited-states are also developed.

math-ph↗
arXiv2003

Asymptotic iteration method for eigenvalue problems

An asymptotic interation method for solving second-order homogeneous linear differential equations of the form y'' = lambda(x) y' + s(x) y is introduced, where lambda(x) \neq 0 and s(x) are C-infinity functions. Applications to Schroedinger type problems, including some with highly singular potentials, are presented.

math-ph↗
arXiv2003

Semiclassical energy formulas for power-law and log potentials in quantum mechanics

We study a single particle which obeys non-relativistic quantum mechanics in R^N and has Hamiltonian H = -Delta + V(r), where V(r) = sgn(q)r^q. If N \geq 2, then q > -2, and if N = 1, then q > -1. The discrete eigenvalues E_{n\ell} may be represented exactly by the semiclassical expression E_{n\ell}(q) = min_{r>0}\{P_{n\ell}(q)^2/r^2+ V(r)}. The case q = 0 corresponds to V(r) = ln(r). By writing one power as a smooth transformation of another, and using envelope theory, it has earlier been proved that the P_{n\ell}(q) functions are monotone increasing. Recent refinements to the comparison theorem of QM in which comparison potentials can cross over, allow us to prove for n = 1 that Q(q)=Z(q)P(q) is monotone increasing, even though the factor Z(q)=(1+q/N)^{1/q} is monotone decreasing. Thus P(q) cannot increase too slowly. This result yields some sharper estimates for power-potential eigenvlaues at the bottom of each angular-momentum subspace.

math-ph↗
arXiv2003

Coulomb plus power-law potentials in quantum mechanics

We study the discrete spectrum of the Hamiltonian H = -Delta + V(r) for the Coulomb plus power-law potential V(r)=-1/r+ beta sgn(q)r^q, where beta > 0, q > -2 and q \ne 0. We show by envelope theory that the discrete eigenvalues E_{n\ell} of H may be approximated by the semiclassical expression E_{n\ell}(q) \approx min_{r>0}\{1/r^2-1/(mu r)+ sgn(q) beta(nu r)^q}. Values of mu and nu are prescribed which yield upper and lower bounds. Accurate upper bounds are also obtained by use of a trial function of the form, psi(r)= r^{\ell+1}e^{-(xr)^{q}}. We give detailed results for V(r) = -1/r + beta r^q, q = 0.5, 1, 2 for n=1, \ell=0,1,2, along with comparison eigenvalues found by direct numerical methods.

math-ph↗
arXiv2003

Relativistic N-Boson Systems Bound by Oscillator Pair Potentials

We study the lowest energy E of a relativistic system of N identical bosons bound by harmonic-oscillator pair potentials in three spatial dimensions. In natural units the system has the semirelativistic ``spinless-Salpeter'' Hamiltonian H = \sum_{i=1}^N \sqrt{m^2 + p_i^2} + \sum_{j>i=1}^N gamma |r_i - r_j|^2, gamma > 0. We derive the following energy bounds: E(N) = min_{r>0} [N (m^2 + 2 (N-1) P^2 / (N r^2))^1/2 + N (N-1) gamma r^2 / 2], N \ge 2, where P=1.376 yields a lower bound and P=3/2 yields an upper bound for all N \ge 2. A sharper lower bound is given by the function P = P(mu), where mu = m(N/(gamma(N-1)^2))^(1/3), which makes the formula for E(2) exact: with this choice of P, the bounds coincide for all N \ge 2 in the Schroedinger limit m --> infinity.

math-ph↗
arXiv2002

Energy bounds for a class of singular potentials and some related series

Perturbation expansions up to third order for the generalized spiked harmonic oscillator Hamiltonians H = -d^2/dx^2+ x^2 + A/x^2 + lambda/x^alpha, A >= 0, 2gamma > alpha, gamma=1+(1/2)sqrt(1+4A), and small values of the coupling lambda > 0, are developed. Upper and lower bounds for the eigenvalues are computed by means of the procedure of Burrows et al [J. Phys. A: Math. Gen. 20, 889-897 (1987)] for assessing the accuracy of a truncated perturbation expansion. Closed-form sums for some related perturbation double infinite series then immediately follow as a result of this investigation.

math-ph↗
arXiv2002

Generalized comparison theorems in quantum mechanics

This paper is concerned with the discrete spectra of Schroedinger operators H = -Delta + V, where V(r) is an attractive potential in N spatial dimensions. Two principal results are reported for the bottom of the spectrum of H in each angular-momentum subspace H_{ell}: (i) an optimized lower bound when the potential is a sum of terms V(r) = V^{(1)}(r) + V^{(2)}(r), and the bottoms of the spectra of -Delta + V^{(1)}(r) and -Delta + V^{(2)}(r) in H_{ell} are known, and (ii) a generalized comparison theorem which predicts spectral ordering when the graphs of the comparison potentials V^{(1)}(r) and V^{(2)}(r) intersect in a controlled way. Pure power-law potentials are studied and an application of the results to the Coulomb-plus-linear potential V(r) = -a/r + br is presented in detail: for this problem an earlier formula for energy bounds is sharpened and generalized to N dimensions.

math-ph↗
arXiv2002

Convexity and potential sums for Salpeter-like Hamiltonians

The semirelativistic Hamiltonian H = β\sqrt{m^2 + p^2} + V(r), where V(r) is a central potential in R^3, is concave in p^2 and convex in p. This fact enables us to obtain complementary energy bounds for the discrete spectrum of H. By extending the notion of 'kinetic potential' we are able to find general energy bounds on the ground-state energy E corresponding to potentials with the form V = sum_{i}a_{i}f^{(i)}(r). In the case of sums of powers and the log potential, where V(r) = sum_{q\ne 0} a(q) sgn(q)r^q + a(0)ln(r), the bounds can all be expressed in the semi-classical form E \approx \min_{r}{β\sqrt{m^2 + 1/r^2} + sum_{q\ne 0} a(q)sgn(q)(rP(q))^q + a(0)ln(rP(0))}. 'Upper' and 'lower' P-numbers are provided for q = -1,1,2, and for the log potential q = 0. Some specific examples are discussed, to show the quality of the bounds.

math-ph↗
arXiv2002

Closed-form sums for some perturbation series involving hypergeometric functions

Infinite series of the type Sum{n=1,infinity}(alpha/2)_n_2F_1(-n, b; gamma; y)/(n n!) are investigated. Closed-form sums are obtained for alpha a positive integer alpha=1,2,3, ... The limiting case of b --> infinity, after y is replaced with x^2/b, leads to Sum{n=1,infinity}(alpha/2)_n_1F_1(-n; gamma; x^2)/(n n!). This type of series appears in the first-order perturbation correction for the wavefunction of the generalized spiked harmonic oscillator Hamiltonian H = -(d/dx)^2 + Bx^2 + A/x^2 + lambda/x^alpha, x >=0, alpha, lambda > 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)^2 + Bx^2 + lambda/x^alpha, x >= 0, alpha, lambda > 0.

math-ph↗
arXiv2002

Spectral bounds for the cutoff Coulomb potential

The method of potential envelopes is used to analyse the bound-state spectrum of the Schroedinger Hamiltonian H = -Delta -v/(r+b), where v and b are positive. We established simple formulas yielding upper and lower energy bounds for all the energy eigenvalues.

math-ph↗
arXiv2001

Closed-form sums for some perturbation series involving associated Laguerre polynomials

Infinite series sum_{n=1}^infty {(alpha/2)_n / (n n!)}_1F_1(-n, gamma, x^2), where_1F_1(-n, gamma, x^2)={n!_(gamma)_n}L_n^(gamma-1)(x^2), appear in the first-order perturbation correction for the wavefunction of the generalized spiked harmonic oscillator Hamiltonian H = -d^2/dx^2 + B x^2 + A/x^2 + lambda/x^alpha 0 <= x < infty, alpha, lambda > 0, A >= 0. It is proved that the series is convergent for all x > 0 and 2 gamma > alpha, where gamma = 1 + (1/2)sqrt(1+4A). Closed-form sums are presented for these series for the cases alpha = 2, 4, and 6. A general formula for finding the sum for alpha/2 = 2 + m, m = 0,1,2, ..., in terms of associated Laguerre polynomials, is also provided.

math-ph↗
arXiv2001

Energy bounds for the spinless Salpeter equation

We study the spectrum of the spinless-Salpeter Hamiltonian H = β\sqrt{m^2 + p^2} + V(r), where V(r) is an attractive central potential in three dimensions. If V(r) is a convex transformation of the Coulomb potential -1/r and a concave transformation of the harmonic-oscillator potential r^2, then upper and lower bounds on the discrete eigenvalues of H can be constructed, which may all be expressed in the form E = min_{r>0} [ β\sqrt{m^2 + P^2/r^2} + V(r) ] for suitable values of P here provided. At the critical point the relative growth to the Coulomb potential h(r)=-1/r must be bounded by dV/dh < 2β/π.

hep-th↗