Abstract. We study a Sturm-Liouville expression with indefinite weight of the form sgn(−d 2 /dx 2 + V ) on R and the non-real eigenvalues of an associated selfadjoint operator in a Krein space. For real-valued potentials V with a certain behaviour at ±∞ we prove that there are no real eigenvalues and that the number of non-real eigenvalues (counting multiplicities) coincides with the number of negative eigenvalues of the selfadjoint operator associated toThe general results are illustrated with examples.

This paper is concerned with the discrete spectra of Schrödinger operators H = −∆ + 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 ℓ : (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 −∆+V (1) (r) and −∆+V (2) (r) in H ℓ 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.

We study the discrete spectrum of the Hamiltonian H = −∆ + V (r) for the Coulomb plus power-law potential V (r) = −1/r + β sgn(q)r q , where β > 0, q > −2 and q = 0 . We show by envelope theory that the discrete eigenvalues E nℓ of H may be approximated by the semiclassical expression E nℓ (q) ≈ min r>0 {1/r 2 − 1/(µr) + sgn(q)β(νr) q }. Values of µ and ν are prescribed which yield upper and lower bounds. Accurate upper bounds are also obtained by use of a trial function of the form, ψ(r) = r ℓ+1 e −(xr) q . We give detailed results for V (r) = −1/r + βr q , q = 0.5, 1, 2 for n = 1, ℓ = 0, 1, 2, along with comparison eigenvalues found by direct numerical methods.

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.Comment: 25 page

We derive a new criterion for deducing the intervals of oscillatory behavior in the solutions of ordinary second-order linear homogeneous differential equations from their coefficients. The validity of the method depends on one's ability to transform a given differential equation to its simplest possible form, so a program must be executed that involves transformations of both variables before the criterion can be applied. The payoff of the program is the detection of oscillations precisely where they may occur in finite or infinite intervals of the independent variable. We demonstrate how the oscillation-detection program can be carried out for a variety of well-known differential equations from applied mathematics and mathematical physics. MSC: 34A25; 34A30