Spectral asymptotics for fourth order differential operator with two turning points

Spectral Estimates for a Periodic Fourth-order Operator

The operator H = d 4 dt4 + d dt p d dt + q with periodic coefficients p, q on the real line is considered. The spectrum of H is absolutely continuous and consists of intervals separated by gaps. The following statements are proved: 1) the endpoints of gaps are periodic or antiperiodic eigenvalues or branch points of the Lyapunov function, and moreover, their asymptotic behavior at high energy i...

Asymptotic Theory of Second Order Differential Equations with Two Simple Turning Points

1. In t roduct ion x. In t93t Langer initiated and gave the first of numerous contributions to what has become a successful theory for asymptotic expansions ~f the solutions of a differential equation with a turning point. This theory has been extended and applied to a great many questions by him and by others. An extensive list of references may be found in L. CESAm'S book, Asymptotic behavior...

On the Solvability of a Fourth Order Operator Differential Equations

We establish sufficient conditions for the solvability of a fourth order operator differential equation. These conditions are expressed in terms of operator coefficients. 1. Preliminaries In this paper we consider the problem P (d/dt)u = du dt4 +Au+ 4 ∑ j=0 Aj d4−ju dt4−j = f, (1.1) e1 ui(0) = 0, i = 0, 1, si ∈ {0, 1, 2, 3}, (1.2) e2 where f ∈ L2(R+,H), and A0, A1, A2, A3, A4 and A are unbounde...

New family of Two-Parameters Iterative Methods for Non-Linear Equations with Fourth-Order Convergence

On the fourth order PI equation and coalescing phenomena of nonlinear turning points

In this report we present a conjecture for the fourth order PI equation with a large parameter to show its importance in the exact WKB analysis. The conjecture is related to coalescing phenomena of turning points and can be regarded as a nonlinear analogue of Hirose’s result ([9]) for the Pearcey system. We also discuss some relations between the conjecture and Dubrovin’s result ([7]) for the K...