Exact Solutions for the Singularly Perturbed Riccati Equation and Exact WKB Analysis

Abstract

The singularly perturbed Riccati equation is the first-order nonlinear ordinary differential equation $\hbar \partial _x f = af^2 + bf + c$ in the complex domain where $\hbar $ is a small complex parameter. We prove an existence and uniqueness theorem for exact solutions with prescribed asymptotics as $\hbar \to 0$ in a half-plane. These exact solutions are constructed using the Borel–Laplace method; that is, they are Borel summations of the formal divergent $\hbar $ -power series solutions. As an application, we prove existence and uniqueness of exact WKB solutions for the complex one-dimensional Schrödinger equation with a rational potential.

Existence and Uniqueness of Exact WKB Solutions for Second-Order Singularly Perturbed Linear ODEs

“…However, although the existence of exact WKB solutions in classes of examples has been established, a general existence theorem for second-order linear ODEs has remained unavailable. Contributions towards such a general theory include Gerard and Grigis [55], Bodine, Dunster, Lutz, and Schäfke [29,56,57], Giller and Milczarski [58], Koike and Takei [59], Ferreira, López, and Sinusía [60,61], as well as most recently by Nemes [62] whose preprint appeared at roughly the same time as our previous work [1] that underpins our results here. Our paper contributes to this long line of work by establishing a general theory of existence and uniqueness of exact WKB solutions, which generalises the relevant results from the aforementioned works (see subsection 5.5 for a discussion).…”

“…Let us also stress that (3) is stronger than the usual notion of Gevrey asymptotics in that we require the above bounds to hold → 0 uniformly in all directions within S (see section 5.5). This stronger asymptotic assumption plays a crucial role in our analysis by allowing us to draw uniqueness conclusions with the help of a theorem of Nevanlinna [77,78] (see Theorem C.1; see also [1,Theorem B.11] where we present a detailed proof).…”

“…In fact, on certain subsets U ⊂ X , this proposition can be seen as a consequence of Theorem 5.1 (or more specifically of Lemma 5.2). In more generality, a direct proof can be found in [1,Lemma 3.11].…”

“…To keep the discussion a little more elementary, we state the relevant definitions and facts by appealing directly to explicit formulas using the Liouville transformation (defined below) commonly used in the WKB analysis of Schrödinger equations. 1 4.1. Résumé.…”

“…These are holomorphic solutions that are canonically specified (in a precise sense via Borel resummation) by their exponential asymptotic expansions as → 0 in a halfplane. They are constructed by means of the Borel-Laplace method for the associated singularly perturbed Riccati equation which we investigated in [1].…”