A Liouville’s Formula for Systems with Reflection

In this work, we derived an Abel–Jacobi–Liouville identity for the case of two-dimensional linear systems of ODEs (ordinary differential equations) with reflection. We also present a conjecture for the general case and an application to coupled harmonic oscillatorsF. Adrián F. Tojo was partially supported by Xunta de Galicia, project ED431C 2019/02, and by the Agencia Estatal de Investigación (AEI) of Spain under grant MTM2016-75140-P, co-financed by the European Community fund FEDERS

DynaStI: A Dynamic Retention Time Database for Steroidomics

Steroidomics studies face the challenge of separating analytical compounds with very similar structures (i.e., isomers). Liquid chromatography (LC) is commonly used to this end, but the shared core structure of this family of compounds compromises effective separations among the numerous chemical analytes with comparable physico-chemical properties. Careful tuning of the mobile phase gradient and an appropriate choice of the stationary phase can be used to overcome this problem, in turn modifying the retention times in different ways for each compound. In the usual workflow, this approach is suboptimal for the annotation of features based on retention times since it requires characterizing a library of known compounds for every fine-tuned configuration. We introduce a software solution, DynaStI, that is capable of annotating liquid chromatography-mass spectrometry (LC–MS) features by dynamically generating the retention times from a database containing intrinsic properties of a library of metabolites. DynaStI uses the well-established linear solvent strength (LSS) model for reversed-phase LC. Given a list of LC–MS features and some characteristics of the LC setup, this software computes the corresponding retention times for the internal database and then annotates the features using the exact masses with predicted retention times at the working conditions. DynaStI is able to automatically calibrate its predictions to compensate for deviations in the input parameters. The database also includes identification and structural information for each annotation, such as IUPAC name, CAS number, SMILES string, metabolic pathways, and links to external metabolomic or lipidomic databases

A geometric approach to non-perturbative quantum mechanics

This work explores the connection between spectral theory and topological strings. A concrete example (the Y(3,0) geometry) of a conjectured exact relation between both based on mirror symmetry (TS/ST correspondence) is analysed in detail, to find a complete agreement. This is used as motivation to apply string theory tools, and in particular the refined holomorphic anomaly, to other Schrödinger-type spectral problems. With it, we efficiently compute their all-orders WKB expansion. We also upgrade the refined holomorphic anomaly to include non-perturbative corrections to the WKB series. This is used to retrieve the transseries generated by previously known exact quantization conditions for quantum mechanical problems. Via resurgence, it will allows us to reproduce the large order behaviour of the WKB coefficients, for both the Schrödinger problems and the quantum mirror curves of the TS/ST correspondence

On the resummation of the Lee-Yang edge singularity coupled to gravity

We study the Borel-Pade resummation of the asymptotic series for the string equation of the Lee-Yang edge singularity. Numerical methods are provided to compute a high accuracy exact solution. We find the resummation matches the numerical integration without need for further non-perturbative corrections

Holomorphic anomaly and quantum mechanics

We show that the all-orders WKB periods of one-dimensional quantum mechanical oscillators are governed by the refined holomorphic anomaly equations of topological string theory. We analyze in detail the double-well potential and the cubic and quartic oscillators, and we calculate the WKB expansion of their quantum free energies by using the direct integration of the anomaly equations. We reproduce in this way all known results about the quantum periods of these models, which we express in terms of modular forms on the WKB curve. As an application of our results, we study the large order behavior of the WKB expansion in the case of the double well, which displays the double factorial growth typical of string theory

Holomorphic Anomaly and Quantum Mechanics

We show that the all-orders WKB periods of one-dimensional quantum mechanical oscillators are governed by the refined holomorphic anomaly equations of topological string theory. We analyze in detail the double-well potential and the cubic and quartic oscillators, and we calculate the WKB expansion of their quantum free energies by using the direct integration of the anomaly equations. We reproduce in this way all known results about the quantum periods of these models, which we express in terms of modular forms on the WKB curve. As an application of our results, we study the large order behavior of the WKB expansion in the case of the double well, which displays the double factorial growth typical of string theory