# Tomorrow's Stokes Webinar: Singularly Perturbed Riccati Equation and the Exact WKB Method

### 30 Mar 2020

The Stokes Webinar: Singularly Perturbed Riccati Equation and the Exact WKB Method, with our member Nikita Nikolaev (UniGE, Alekseev Group) will be taking place tomorrow 31st March 2020 at 15:00-17:00 (CET).

DATE: 31 March 2020 (Tuesday)

LOCAL TIME: 15:00-17:00 (CET)

VIRTUAL LOCATION:

Zoom Meeting ID: 512-796-617

Join URL: https://unige.zoom.us/j/512796617

**SPEAKER: **Nikita Nikolaev (Geneva)

**TITLE:** Singularly Perturbed Riccati Equation and the Exact WKB Method

**ABSTRACT:**

A singularly perturbed Riccati equation is the differential equation of the form \( \hbar \partial_x f = a f^2 + bf + c \), which is the simplest first-order nonlinear differential equation. The parameter \( \hbar \) is assumed to be small, rendering this a problem in singular perturbation theory. We consider this problem in the complex domain (i.e., both \( x \) and \( \hbar \) are complex), and the coefficients \( a, b, c \) are meromorphic functions of \((x,\hbar)\) with prescribed asymptotic expansions as \( \hbar \to 0 \) in some sector.

The Riccati equation appears in a wide variety of contexts, from optimal control theory, to dynamic programming, to projective differential geometry, to quantum mechanics, to algebraic geometry of meromorphic connections, and the list goes on. The latter two subjects are the primary sources of motivation for me. Notably, if you consider the one-dimensional time-independent Schrodinger equation, the famous WKB method for finding approximate solutions with prescribed asymptotic behaviour as \( \hbar \to 0 \) boils down to finding formal solutions to a singularly perturbed Riccati equation. This very powerful approximation method is at least as old quantum mechanics itself. However, the natural and difficult question of finding exact solutions with prescribed asymptotic behaviour as \( \hbar \to 0 \) has remained open in general. Such solutions are often dubbed exact WKB solutions, and the goal of constructing them is at the heart of what is known as the exact WKB method.

Existence and uniqueness theory for the Riccati equation (and indeed for much more general classes of equations) is a very classical and very well developed subject in differential equations. However, existence and uniqueness of solutions with prescribed asymptotics in the perturbation parameter \( \hbar \) is not understood in general. I will explain how I managed to fill in this gap by using some powerful modern techniques from the theory of resurgent asymptotic analysis. As the most important application, I am able to deduce existence and uniqueness of exact WKB solutions.

This talk is partially based on my preprint arXiv:1909.04011 and some work in preparation.

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**LOCAL TIME:** 15:00-17:00 CET (Geneva, Paris, Berlin, Madrid, Rome, Amsterdam, Brussels)

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