Related papers: WKB solutions of difference equations and reconstr…
We introduce a geometric refinement of Gromov-Witten invariants for $\mathbb P^1$-bundles relative to the natural fiberwise boundary structure. We call these refined invariant correlated Gromov-Witten invariants. Furthermore, we prove a…
We study properties of the recently established refined topological recursion for some simple spectral curves associated to quadratic differentials. We prove explicit formulas for the free energy and Voros coefficients of the corresponding…
A construction of general solutions of the \hbar-dependent Toda hierarchy is presented. The construction is based on a Riemann-Hilbert problem for the pairs (L,M) and (\bar L,\bar M) of Lax and Orlov-Schulman operators. This Riemann-Hilbert…
The goal of this article is to rederive the connection between the Painlev\'e $5$ integrable system and the universal eigenvalues correlation functions of double-scaled hermitian matrix models, through the topological recursion method. More…
We define higher quantum Airy structures as generalizations of the Kontsevich-Soibelman quantum Airy structures by allowing differential operators of arbitrary order (instead of only quadratic). We construct many classes of examples of…
The connection between the method of comparison equations (generalized WKB method) and the Ermakov-Pinney equation is established. A perturbative scheme of solution of the generalized Ermakov-Pinney equation is developed and is applied to…
We study the non-perturbative quantum geometry of the open and closed topological string on the resolved conifold and its mirror. Our tools are finite difference equations in the open and closed string moduli and the resurgence analysis of…
We study the correlators $W_{g,n}$ arising from Orlov-Scherbin 2-Toda tau functions with rational content-weight $G(z)$, at arbitrary values of the two sets of time parameters. Combinatorially, they correspond to generating functions of…
We first review our previous work arxiv:1503.02993 [math-ph] where we considered a model for topological recursion based on the Hopf Algebra of planar binary trees of Loday and Ronco and showed that extending this Hopf Algebra by…
Kontsevich-Soibelman (2017) reformulated Eynard-Orantin topological recursion (2007) in terms of Airy structure which provides some geometrical insights into the relationship between the moduli space of curves and topological recursion. In…
In this article, we investigate the resurgent properties of the WKB solutions for a singularly perturbated second order ordinary differential equation. In particular, we extend and propose a new proof of a theorem due to Aoki (et al) near a…
Multiparametric families of hypergeometric $\tau$-functions of KP or Toda type serve as generating functions for weighted Hurwitz numbers, providing weighted enumerations of branched covers of the Riemann sphere. A graphical interpretation…
This is a survey article describing the relationship between quantum curves and topological recursion. A quantum curve is a Schr\"odinger operator-like noncommutative analogue of a plane curve which encodes (quantum) enumerative invariants…
In this paper, we use the WKB approximation method to approximately solve a deformed Schrodinger-like differential equation: $\left[ -\hbar^{2} \partial_{\xi}^{2}g^{2}\left( -i\hbar\alpha\partial_{\xi}\right) -p^{2}\left( \xi\right) \right]…
In the framework of toroidal Pseudodifferential operators on the flat torus $\Bbb T^n := (\Bbb R / 2\pi \Bbb Z)^n$ we begin by proving the closure under composition for the class of Weyl operators $\mathrm{Op}^w_\hbar(b)$ with simbols $b…
We propose an extension of Wenzel-Kramers-Brillouin (WKB) approximation for solving the Schr\"odinger equation. A set of coupled differential equations is obtained by considering an ansatz of the wave function with an auxiliary condition on…
We obtain the rigorous WKB expansion to all orders for the radial Kepler problem, using the residue calculus in evaluating the WKB quantization condition in terms of a complex contour integral in the complexified coordinate plane. The…
We consider the difference Schr{\"o}dinger equation $\psi$(z + h) + $\psi$(z -- h) + v(z)$\psi$(z) = 0 where z is a complex variable, h > 0 is a parameter, and v is an analytic function. As h $\rightarrow$ 0 analytic solutions to this…
In this paper, we show that it is always possible to deform a differential equation $\partial_x \Psi(x) = L(x) \Psi(x)$ with $L(x) \in \mathfrak{sl}_2(\mathbb{C})(x)$ by introducing a small formal parameter $\hbar$ in such a way that it…
We prove that formal WKB solutions of Schr\"odinger equations on Riemann surfaces are resurgent. Specifically, they are Borel summable in almost all directions and their Borel transforms admit endless analytic continuation away from a…