The double points of Mathieu’s differential equation

Abstract

Abstract. Mathieu's differential equation, y" + (a -2q cos 2x)y = 0, admits of solutions of period ir or 2x for four countable sets of characteristic values, aiq), which can be ordered as ariq), r = 0, 1, • • -. The power series expansions for the ariq) converge up to the first double point for that order in the complex plane. [At a double point, ar(g) = ar+2iq).] The present work furnishes the double points for orders r up to and including 15. These double points are singular points, and the usual methods of determining the characteristic values break down at a singular point. However, it was possible to determine two smooth functions in which one could interpolate for both q and aTiq) at the singular point. The method is quite general and can be used in other problems as well. |

Simple and double eigenvalues of the Hill operator with a two-term potential

“…Again, L P er + , or its restriction K odd , has a Jordan block. 5. Examples in the previous subsection show that in Lemmas 9, 10 and Theorem 11 the assumptions that a, b be real, or L be self-adjoint, are important.…”

“…Examples in the previous subsection show that in Lemmas 9, 10 and Theorem 11 the assumptions that a, b be real, or L be self-adjoint, are important. But let us follow [3,35,19,4,5,37] and raise a general question about the structure of spectral Riemann surfaces related to these problems. Of course, it would be interesting to change both and t in complex plane, i.e., to consider ( , t) ∈ C 2 but for a while, let us talk about fixed positive t. Define, for each t > 0, four surfaces .17), and H ± are defined by (3.44).…”

Trace formula and Spectral Riemann Surfaces for a class of tri-diagonal matrices

“…by Bender and Wu [1]; see also [25][26][27][28]34,35]. The case of Mathieu-Hill operators could be deduced to Example (1.3) + (1.4); it has a longer history (see [2,3,10,19,20,37,39,40]). Some observations about Whittaker-Hill operators could be found in [5,Section 5.4].…”

“…On the other hand, (4.18) and (4.19) imply that where C n = (8MR) 14 k 0 (8MR) 2k /n k is a bounded sequence. On the other hand, from (4.18) to (4.23), it follows that 3 (4.48)…”

Zum Problem der Irreduzibilität für zweiparametrige Eigenwertaufgaben.

“…Gewissen Aufschluß gaben hier numerisch-funktionentheoretische Untersuchungen mit elektronischen Rechenanlagen, über die wir in [2], [3] zum Teil berichtet haben, und die inzwischen auch G. Blanch und D. S. Clemm, [4], durchgeführt haben. Sie zeigen, daß die ersten a 2n sämtlich Zweige derselben Funktion sind.…”