David B. Pearson

In recent years, novel quantifications of measurement error in quantum mechanics have for the first time enabled precise formulations of Heisenberg's famous but often challenged measurement uncertainty relation. These relations take the form of a trade-off for the necessary errors in joint approximate measurements of position and momentum and other incompatible pairs of observables. Here we review some of these error measures, examine their properties and suitability, and compare their relative strengths as criteria for "good" approximations.

In this paper, we consider the Sturm–Liouville equation −y′′ + qy = λy on the half line (0, ∞) under the assumptions that x=0 is a regular singular point and nonoscillatory for all real λ, and that either (i) q is L₁ near x=∞, or (ii) q′ is L₁ near ∞ with q(x) → 0 as x → ∞, so that there is absolutely continuous spectrum in (0, ∞). Characterizations of the spectral density function for this doubly singular problem, similar to those obtained in Fulton et al. (2008, J. Comp. Appl. Math., 212, 194–213) and Fulton et al. (2008, J. Comp. Appl. Math., 212, 150–178) (when the left endpoint is regular) are established; corresponding approximants from the two algorithms in these papers are then utilized, along with Frobenius recurrence relations and piecewise trigonometric/hyperbolic splines, to generate numerical approximations to the spectral density function associated with the doubly singular problem on (0, ∞). In the case of the radial part of the separated hydrogen atom problem, the new algorithms are capable of achieving near machine precision accuracy over the range of λ from 0 to 10000, accuracies which could not be achieved using the SLEDGE software package.

We formulate and prove a new, universally valid uncertainty relation for the necessary errors bar widths in any approximate joint measurement of position and momentum.

We apply the methods of value distribution theory to the spectral asymptotics of Schrodinger operators with L^2-sparse potentials.