Spectral bounds and comparison theorems for Schrödinger operators

One of the most important problems in quantum physics is to find the energy eigenvalues for Schrödinger's equation. This equation is exactly solvable only for a small class of potentials. For one-particle problems numerical solutions can always be obtained, but in the absence of exact solutions, the next best thing is an analytical formula for an approximation, such as energy bound. In this thesis we use geometrical techniques such as the envelope method to obtain analytical spectral bounds for Schrödinger's equation for wide classes of potential. Our geometrical approach leans heavily on the comparison theorem, to the effect that V 1 < V 2 [implies] E 1 < E 2 . For the bottom of an angular-momentum subspace it is possible to generalize the comparison theorem by allowing the comparison potentials V 1 and V 2 to cross over in a controlled way and still imply spectral ordering E 1 < E 2 . We prove and use these theorems to sharper some earlier upper and lower bounds obtained using the 'envelope method'. In chapter two we introduce the envelope method that is used in the subsequent chapters. In chapter three we study the Hellmann potential in quantum physics: we prove that discrete eigenvalues exist and we obtain formulae for upper and lower bounds to them. In chapter four, we prove the existence of a discrete spectrum for the cutoff-Coulomb potential, and we obtain upper and lower-bound formulae. In chapter five we prove the monotonicity of the wave function |( r ) for the ground state in the case of attractive central potentials in N N spatial dimensions. By using this result we establish some generalized comparison theorems in which the comparison potentials intersect. We use these theorems, together with the sum approximation, to improve the upper and the lower bounds obtained earlier with the aid of the envelope method. In chapter six, we study the representation P ( q ) for the eigenvalues E ( q ) of the operator H = -x + sgn(q)r q defined by E ( q ) = [Special characters omitted.] . It had earlier been proved that P ( q ) is monotone increasing. We strengthen this result for the ground state (and the bottom of each angular-momentum subspace) by using the generalized comparison theorems to prove that a new function Q ( q ) = Z ( q ) P ( q ) is monotone increasing, where the factor Z ( q ) is monotone decreasing. Thus we know that P ( q ) cannot increase too slowly: this in turn allows us to obtain same improved bounds for the eigenvalues E ( q ) in N dimensions. In the last chapter we analyse bounds for the Coulomb plus power-law potentials obtained by variational methods and the sum approximation

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.Comment: 11 pages, 2 figure

Spectral bounds for the Hellmann potential

The method of potential envelopes is used to analyse the bound state spectrum of the Schroedinger Hamiltonian H=-\Delta+V(r), where the Hellmann potential is given by V(r) = -A/r + Be^{-Cr}/r, A and C are positive, and B can be positive or negative. We established simple formulas yielding upper and lower bounds for all the energy eigenvalues.Comment: 9 pages, 2 figures, typos correcte

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$, then the eigenvalues E = E_{n\ell}^{(d)}(\lambda) are given approximately by the semi-classical expression E = \min_{r > 0}[\frac{1}{r^2} + \sum_{i = 1}^{k}a_i(P_ir)^{q_i}]. It is proved that this formula yields a lower bound if P_i = P_{n\ell}^{(d)}(q_1), an upper bound if$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+\lambda r^{2m},m=2,3,... in d dimension, for example, E = E_{n\ell}^{(d)}(\lambda) is determined by the algebraic expression \lambda={1\over \beta}({2\alpha(m-1)\over mE-\delta})^m({4\alpha \over (mE-\delta)}-{E\over (m-1)}) where \delta={\sqrt{E^2m^2-4\alpha(m^2-1)}} and \alpha, \beta are 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.Comment: 13 pages, no figure

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.Comment: 11 pages, 3 figure

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.Comment: 20 pages, 5 figure

A basis for variational calculations in d dimensions

In this paper we derive expressions for matrix elements (\phi_i,H\phi_j) for the Hamiltonian H=-\Delta+\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.Comment: 22 page