Algebraic Analysis of Singular Perturbation Theory

Quantum geometry of resurgent perturbative/nonperturbative relations

“…To address this question, we first translate the previous results for theseÁlvarez-Casares-type perturbative/non-perturbative (P/NP) relations [17-23, 25, 26] into a more geometric language, better suited for a path integral formulation. We adopt the semiclassical path integral approach of Balian and Bloch [34,35] and the geometric Stokes diagram and monodromy picture [6][7][8]36], to express the P/NP relations in terms of all-orders WKB ("exact WKB") actions and dual actions. In so doing, it proves useful to adopt some of the language and ideas from supersymmetric gauge theories, integrability, conformal field theory and wall-crossing [37][38][39][40][41][42][43][44][45][46][47][48], given the close connection of such theories with exact WKB [49][50][51][52][53][54][55][56][57][58][59][60][61][62][63][64][65].…”

“…The general result now follows because the energy spectrum is determined by an exact quantization condition which is a monodromy condition expressed in terms of a(u, ) and a D (u, ) [6][7][8]36]. Physically, we can think of the cycle α as describing the usual Bohr-Sommerfeld perturbative physics, while the other cycle β describes the non-perturbative tunneling physics.…”

“…Furthermore, using the classical Picard-Fuchs equation (5.19), (5.20), we can 8 In fact an improved WKB ansatz ψ = exp i x φ(x , u; )dx / √ φ automatically resums these odd terms and generates a Riccati equation that depends on 2 . The recursion relations of this improved Riccati equation directly give the even terms in (5.29).…”

“…Two distinct types of resurgent behavior have been identified in quantum spectral problems. The first is a generic type of "large order/low order" form of resurgence [6][7][8][9][10][11][12], whereby the large order growth of the perturbative coefficients of fluctuations about a given non-perturbative sector is related to the low-order perturbative coefficients of fluctuations about other non-perturbative sectors. This resurgent structure encodes an intricate network of relations between different non-perturbative sectors, and reflects to a surprising degree the generic resurgent structure of the all-orders steepest descents analysis of ordinary exponential integrals [13,14], and indeed the general resurgent structure of real trans-series [15].…”

On the multisummability of WKB solutions of certain singularly perturbed linear ordinary differential equations

“…is successfully analyzed in an explicit manner by using Borel resummed WKB solutions. (See, for example, [3,5].) For the Borel summability of WKB solutions of (1.5) we refer the readers to [2,4,6] and references cited there.…”