We study convergence of solutions of a space and time inhomogeneous fractional wave equation on the quarter-plane to the stationary regime described by solutions of the Helmholtz equation.

We study the ramified Cauchy problem for a linear PDE with a radial point using the theory of microdifferential operators.

We re-examine Shatalov-Sternin's proof of existence of resurgent solutions of a linear ODE. In particular, we take a closer look at the "Riemann surface" (actually, a two-dimensional complex manifold) whose existence, endless continuability and other properties are claimed by those authors. We present a detailed argument for a part of the "Riemann surface" relevant for the exact WKB method. The present text is the author's article arXiv:0907.2934 rewritten from a different perspective.

In the context of complex WKB analysis, we discuss a one-dimensional Schrödinger equation−h2∂x2f(x,h)+[Q(x)+hQ1(x,h)]f(x,h)=0,h→0, where Q(x), Q1(x,h) are analytic near the origin x=0, Q(0)=0, and Q1(x,h) is a factorially divergent power series in h. We show that there is a change of independent variable y=y(x,h), analytic near x=0 and factorially divergent with respect to h, that transforms the above Schrödinger equation to a canonical form. The proof goes by reduction to a mildly nonlinear equation on y(x,h) and by solving it using an appropriately modified Newton's method of tangents.

Kashiwara-Schapira style sheaf theory is used to justify analytic continuability of solutions of a Laplace transformed Schroedinger equation with a small parameter. This partially proves the description of the Stokes phenomenon for WKB asymptotics predicted by Voros in 1983.

The Witten Laplacian corresponding to a Morse function on the circle is studied using methods of complex WKB and resurgent analysis. It is shown that under certain assumptions the low-lying eigenvalues of the Witten Laplacian are resurgent.

The paper is devoted to some foundational questions in resurgent analysis. As a main technical result, it is shown that under appropriate conditions the infinite sum of endlessly continuable majors commutes with the Laplace transform. A similar statement is proven for compatibility of a convolution and of an infinite sum of majors. We generalize the results of Candelpergher-Nosmas-Pham and prove a theorem about substitution of a small (extended) resurgent function into a holomorphic parameter of another resurgent function. Finally, we discuss an application of these results to the question of resurgence of eigenfunctions of a one-dimensional Schrodinger operator corresponding to a small resurgent eigenvalue.

We re-examine Shatalov-Sternin's proof of existence of resurgent solutions of a linear ODE. In particular, we take a closer look at the "Riemann surface" (actually, a two-dimensional complex manifold) whose existence, endless continuability and other properties are claimed by those authors. We present a detailed argument for a part of the "Riemann surface" most relevant for the exact WKB method.

Families of translates and homothets of strictly convex curves are proven to possess Helly-type properties generalizing those of a circle. Weaker results are shown for arbitrary convex curves.

Mathematical instanton bundles of rank 4 and c_2=2 on ℙ^4 have a smoothquasiprojective moduli space, which is shown via a direct GIT construction. A complete classification of jumping lines of these vector bundles is obtained. The duals of these bundles are described. Dimension computations and irreducibility proofs for some "interesting" subsets of the moduli space are presented. Restrictions of vector bundles from ℙ^4 to ℙ^3 are shown to produce mathematical instanton bundles on ℙ^3.