Large scale radial stability density of Hill's equation

Abstract

This paper deals with large scale aspects of Hill's equationẍ +(a +bp(t))x = 0, where p is periodic with a fixed period. In particular, the interest is the asymptotic radial density of the stability domain in the (a, b)-plane. It turns out that this density changes discontinuously in a certain direction and exhibits and interesting asymptotic fine structure. Most of the paper deals with the case where p is a Morse function with one maximum and one minimum, but also the discontinuous case of square Hill's equation is studied, where the density behaves differently.Mathematics Subject Classification: 34C08, 34D23, 26B15, 54C30

Two-frequency operation of a Paul trap to optimise confinement of two species of ions

“…For q (τ n ) A = 0.4 and n = 100 we find that a eff = 800. This is far greater than is usually considered in the theory of Paul traps but the asymptotic properties of the Mathieu equation are known in literature [41,28]. For a, q 1 the critical line between predominantly stable and unstable regions is a = 2q, as shown in Fig.…”

“…For n 10 the stability regions show universal behaviour: a quadratic increase up to V n /2V A 0.7 followed by a linear decrease to zero at V n /2V A = 0.91. The critical curve that traverses equal distances in the stable and unstable regions [28,25] over the range 0 < V n /2V A < 0.71 is a quadratic curve of the form:…”

Cotrapping different species in ion traps using multiple radio frequencies

“…2a, when there is a large number of parametric resonances the fine features of the stability diagram are lost. To differentiate between stable and unstable regions we introduce the concept of a critical line which for the undamped Mathieu equation is simply a = 2q [19][20][21]. The critical line is defined as the geometric collection of points that separate the stability diagram in regions of equal stable/unstable density.…”