arXiv subjects
Publications and source records attributed to Van Higgs.
A linear quantum harmonic oscillator factors into one dimensional oscillators and can be solved using creation and annihilation operators. We consider a spherical analogue. This analogue does not factor. The two dimensional case is critical, and we compute the spectrum and partition function. This is of interest because the spherical oscillator is potentially relevant to chiral models in 2d quantum field theory.
In reference to Werner's measure on self-avoiding loops on Riemann surfaces, Cardy conjectured a formula for the measure of all homotopically nontrivial loops in a finite type annular region with modular parameter $\rho$. Ang, Remy and Sun have announced a proof of this conjecture using random conformal geometry. Cardy's formula implies that the measure of the set of homotopically nontrivial loops in the punctured plane which intersect $S^1$ equals $\frac{2\pi}{\sqrt{3}}$. This set is the disjoint union of the set of loops which avoid a ray from the unit circle to infinity and its complement. There is an inclusion/exclusion sum which, in a limit, calculates the measure of the set of loops which avoid a ray. Each term in the sum involves finding the transfinite diameter of a slit domain. This is numerically accessible using the remarkable Schwarz-Christoffel package developed by Driscoll and Trefethen. Our calculations suggest this sum is around $\pi$, consistent with Cardy's formula.