arXiv subjects

Davide Masoero

Publications and source records attributed to Davide Masoero.

At least 19 recordsLinked to original sources

arXiv2026

ODE/IM Correspondence at the Free-Fermion Point. Laguerre Wronskians, Shifted Symmetric Functions, and Quantum KdV

We consider the ODE/IM correspondence for the value $c=-2$ of the Virasoro central charge (free-fermion point) and the associated quantum KdV model $-$ the quantization of the second hamiltonian structure of the classical periodic KdV model. We prove that the ODE/IM correspondence is complete (in the sense of V. Bazhanov, S. Lukyanov, and A. Zamolodchikov), namely that any solution of the Bethe equations coincides with the spectrum of a rational extension of the (quantum) harmonic oscillator. To this end, on the ODE side we consider Crum$-$Darboux transformations of the harmonic oscillator and the associated Laguerre Wronskians, which are remarkable special functions parametrized by pairs of partitions which we study in depth. As a further result, on the IM side, we diagonalize explicitly the first three hamiltonian operators of quantum KdV (in the free field representation): the eigenstates are Schur functions and the eigenvalues are shifted symmetric functions on partitions. We give two applications of this result: i) we prove that the eigenvalues are given by the evaluation of the Newton symmetric polynomials at the poles of the associated monster potentials, as further conjectured by V. Bazhanov, S. Lukyanov, and A. Zamolodchikov; ii) we show that these hamiltonian operators also belong to the algebra of hamiltonian operators obtained by quantizing the first hamiltonian structure of the classical periodic dispersionless KdV model.

math-ph↗

arXiv2025

Asymptotic solutions for linear ODEs with not-necessarily meromorphic coefficients: a Levinson type theorem on complex domains, and applications

In this paper, we consider systems of linear ordinary differential equations, with analytic coefficients on big sectorial domains, which are asymptotically diagonal for large values of $|z|$. Inspired by N. Levinson's work [Lev48], we introduce two conditions on the dominant diagonal term (the $L$-$condition$) and on the perturbation term (the $good\,\,decay\,\,condition$) of the coefficients of the system, respectively. Under these conditions, we show the existence and uniqueness, on big sectorial domains, of an $asymptotic$ fundamental matrix solution, i.e. asymptotically equivalent (for large $|z|$) to a fundamental system of solutions of the unperturbed diagonal system. Moreover, a refinement (in the case of subdominant solutions) and a generalization (in the case of systems depending on parameters) of this result are given. As a first application, we address the study of a class of ODEs with not-necessarily meromorphic coefficients. We provide sufficient conditions on the coefficients ensuring the existence and uniqueness of an asymptotic fundamental system of solutions, and we give an explicit description of the maximal sectors of validity for such an asymptotics. Furthermore, we also focus on distinguished examples in this class of ODEs arising in the context of open conjectures in Mathematical Physics relating Integrable Quantum Field Theories and affine opers ($ODE/IM\,\,correspondence$). Our results fill two significant gaps in the mathematical literature pertaining to these conjectural relations. As a second application, we consider the classical case of ODEs with meromorphic coefficients. Under an $adequateness$ condition on the coefficients, we show that our results reproduce (with a shorter proof) the main asymptotic existence theorems of Y. Sibuya [Sib62, Sib68] and W. Wasow [Was65] in their optimal refinements.

math.CA↗

arXiv2025

A primer of the complex WKB method, with application to the ODE/IM correspondence

In these lectures, we provide an introduction to the complex WKB method, using as a guiding example a class of anharmonic oscillators that appears in the ODE/IM correspondence. In the first three lectures, we introduce the main objects of the method, such as the WKB function, the integral equations of Volterra type, the quadratic differential and its horizontal/Stokes lines, the Stokes phenomenon, the notion of asymptotic values, the Fock-Goncharov coordinates and their WKB approximation. In the fourth and last lecture, we compute (and prove) the asymptotic behaviour of the spectrum of the anharmonic oscillators in two asymptotic regimes, when the momentum is fixed and the energy is large, and when the momentum (hence also the energy) is large.

math-ph↗

arXiv2024

$Q$-functions for lambda opers

We consider the Schrödinger operators which are constructed from the $λ$-opers corresponding to solutions of the $\widehat{\mathfrak{sl}}_2$ Gaudin Bethe Ansatz equations. We define and study the connection coefficients called the $Q$-functions. We conjecture that the $Q$-functions obtained from the $λ$-opers coincide with the $Q$-functions of the Bazhanov-Lukyanov-Zamolodchikov opers with the monster potential related to the quantum KdV flows. We give supporting evidence for this conjecture.

math-ph↗

arXiv2023

Feigin-Frenkel-Hernandez Opers and the QQ-system

This paper represents the completion of our work on the ODE/IM correspondence for the generalised quantum Drinfeld-Sokolov models. We present a unified and general mathematical theory, encompassing all particular cases that we had already addressed, and we fill important analytic and algebraic gaps in the literature on the ODE/IM correspondence. For every affine Lie algebra $\mathfrak{g}$ -- whose Langlands dual $\mathfrak{g}'$ is the untwisted affinisation of a simple Lie algebra -- we study a class of affine twisted parabolic Miura $\mathfrak{g}$-opers, introduced by Feigin, Frenkel and Hernandez. The Feigin-Frenkel-Hernandez opers are defined by fixing the singularity structure at $0$ and $\infty$, and by allowing a finite number of additional singular terms with trivial monodromy. We define the central connection matrix and Stokes matrix for these opers, and prove that the coefficients of the former satisfy the the $QQ$ system of the quantum $\mathfrak{g}'$-Drinfeld-Sokolov (or quantum $\mathfrak{g}'$-KdV) model. If $\mathfrak{g}$ is untwisted, it is known that the trivial monodromy conditions are equivalent to a complete system of algebraic equations for the additional singularities. We prove a suprising negative result in the case $\mathfrak{g}$ is twisted: in this case, the trivial monodromy conditions have no non-trivial solutions.

math-ph↗

arXiv2023

On solutions of the Bethe Ansatz for the Quantum KdV model

We study the Bethe Ansatz Equations for the Quantum KdV model, which are also known to be solved by the spectral determinants of a specific family of anharmonic oscillators called monster potentials (ODE/IM correspondence). These Bethe Ansatz Equations depend on two parameters, identified with the momentum and the degree at infinity of the anharmonic oscillators. We provide a complete classification of the solutions with only real and positive roots -- when the degree is greater than 2 -- in terms of admissible sequences of holes. In particular, we prove that admissible sequences of holes are naturally parameterised by integer partitions, and we prove that they are in one-to-one correspondence with solutions of the Bethe Ansatz Equations if the momentum is large enough. Consequently, we deduce that the monster potentials are complete, in the sense that every solution of the Bethe Ansatz Equations coincides with the spectrum of a unique monster potential. This essentially (i.e. up to gaps in the previous literature) proves the ODE/IM correspondence for the Quantum KdV model/monster potentials -- which was conjectured by Dorey-Tateo and Bazhanov-Lukyanov-Zamolodchikov -- when the degree is greater than 2. Our approach is based on the transformation of the Bethe Ansatz Equations into a free-boundary nonlinear integral equation -- akin to the equations known in the physics literature as DDV or KBP or NLIE -- of which we develop the mathematical theory from the beginning.

math-ph↗

arXiv2021

Roots of generalised Hermite polynomials when both parameters are large

We study the roots of the generalised Hermite polynomials $H_{m,n}$ when both $m$ and $n$ are large. We prove that the roots, when appropriately rescaled, densely fill a bounded quadrilateral region, called the elliptic region, and organise themselves on a deformed rectangular lattice, as was numerically observed by Clarkson. We describe the elliptic region and the deformed lattice in terms of elliptic integrals and their degenerations. Keywords: Generalised Hermite polynomials; roots asymptotics; Painleve IV; Boutroux Curves; Tritronquee solution.

math.CA↗

arXiv2020

Counting monster potentials

We study the large momentum limit of the monster potentials of Bazhanov-Lukyanov-Zamolodchikov, which -- according to the ODE/IM correspondence -- should correspond to excited states of the Quantum KdV model. We prove that the poles of these potentials asymptotically condensate about the complex equilibria of the ground state potential, and we express the leading correction to such asymptotics in terms of the roots of Wronskians of Hermite polynomials. This allows us to associate to each partition of $N$ a unique monster potential with $N$ roots, of which we compute the spectrum. As a consequence, we prove -- up to a few mathematical technicalities -- that, fixed an integer $N$, the number of monster potentials with $N$ roots coincides with the number of integer partitions of $N$, which is the dimension of the level $N$ subspace of the quantum KdV model. In striking accordance with the ODE/IM correspondence.

math-ph↗

arXiv2020

On the monodromy of the deformed cubic oscillator

We study a second-order linear differential equation known as the deformed cubic oscillator, whose isomonodromic deformations are controlled by the first Painlev{é} equation. We use the generalised monodromy map for this equation to give solutions to the infinite-dimensional Riemann-Hilbert problems arising from the Donaldson-Thomas theory of the A2 quiver. These are the first known solutions to such problems beyond the uncoupled case. The appendix by Davide Masoero contains a WKB analysis of the asymptotics of the monodromy map.

math.CA↗

arXiv2019

Opers for higher states of the quantum Boussinesq model

We study the ODE/IM correspondence for all the states of the quantum Boussinesq model. We consider a particular class of third order linear ordinary differential operators and show that the generalised monodromy data of such operators provide solutions to the Bethe Ansatz equations of the Quantum Boussinesq model.

math-ph↗

arXiv2018

Opers for higher states of quantum KdV models

We study the ODE/IM correspondence for all states of the quantum $\widehat{\mathfrak{g}}$-KdV model, where $\widehat{\mathfrak{g}}$ is the affinization of a simply-laced simple Lie algebra $\mathfrak{g}$. We construct quantum $\widehat{\mathfrak{g}}$-KdV opers as an explicit realization of the class of opers introduced by Feigin and Frenkel, which are defined by fixing the singularity structure at $0$ and $\infty$, and by allowing a finite number of additional singular terms with trivial monodromy. We prove that the generalized monodromy data of the quantum $\widehat{\mathfrak{g}}$-KdV opers satisfy the Bethe Ansatz equations of the quantum $\widehat{\mathfrak{g}}$-KdV model. The trivial monodromy conditions are equivalent to a complete system of algebraic equations for the additional singularities.

math-ph↗

arXiv2018

Poles of Painlevé IV Rationals and their Distribution

We study the distribution of singularities (poles and zeros) of rational solutions of the Painlevé IV equation by means of the isomonodromic deformation method. Singularities are expressed in terms of the roots of generalised Hermite $H_{m,n}$ and generalised Okamoto $Q_{m,n}$ polynomials. We show that roots of generalised Hermite and Okamoto polynomials are described by an inverse monodromy problem for an anharmonic oscillator of degree two. As a consequence they turn out to be classified by the monodromy representation of a class of meromorphic functions with a finite number of singularities introduced by Nevanlinna. We compute the asymptotic distribution of roots of the generalised Hermite polynomials in the asymptotic regime when $m$ is large and $n$ fixed.

math.CA↗

arXiv2016

Asymptotic analysis of noisy fitness maximization, applied to metabolism and growth

We consider a population dynamics model coupling cell growth to a diffusion in the space of metabolic phenotypes as it can be obtained from realistic constraints-based modelling. In the asymptotic regime of slow diffusion, that coincides with the relevant experimental range, the resulting non-linear Fokker-Planck equation is solved for the steady state in the WKB approximation that maps it into the ground state of a quantum particle in an Airy potential plus a centrifugal term. We retrieve scaling laws for growth rate fluctuations and time response with respect to the distance from the maximum growth rate suggesting that suboptimal populations can have a faster response to perturbations.

q-bio.PE↗

arXiv2016

Bethe Ansatz and the Spectral Theory of affine Lie algebra-valued connections I. The simply-laced case

We study the ODE/IM correspondence for ODE associated to $\hat{\mathfrak g}$-valued connections, for a simply-laced Lie algebra $\mathfrak g$. We prove that subdominant solutions to the ODE defined in different fundamental representations satisfy a set of quadratic equations called $Ψ$-system. This allows us to show that the generalized spectral determinants satisfy the Bethe Ansatz equations.

math-ph↗

arXiv2016

Bethe Ansatz and the Spectral Theory of affine Lie algebra--valued connections II. The non simply--laced case

We assess the ODE/IM correspondence for the quantum $\mathfrak{g}$-KdV model, for a non-simply laced Lie algebra $\mathfrak{g}$. This is done by studying a meromorphic connection with values in the Langlands dual algebra of the affine Lie algebra ${\mathfrak{g}}^{(1)}$, and constructing the relevant $Ψ$-system among subdominant solutions. We then use the $Ψ$-system to prove that the generalized spectral determinants satisfy the Bethe Ansatz equations of the quantum $\mathfrak{g}$-KdV model. We also consider generalized Airy functions for twisted Kac--Moody algebras and we construct new explicit solutions to the Bethe Ansatz equations. The paper is a continuation of our previous work on the ODE/IM correspondence for simply-laced Lie algebras.

math-ph↗

arXiv2015

A Laplace's method for series and the semiclassical analysis of epidemiological models

We develop a Laplace's method to compute the asymptotic expansions of sums of sharply peaked sequences. These series arise as discretizations (Riemann sums) of sharply-peaked integrals, whose asymptotic behavior can be computed by the standard Laplace's method. We apply the Laplace's method for series to the WKB (i.e. semiclassical) analysis of stochastic models of population biology, with special focus on the SIS model. In particular we show that two different and widely-used approaches to the semiclassical limit, i.e. either considering a semiclassical probability distribution or a semiclassical generating function, are equivalent.

math-ph↗

arXiv2015

Critical behaviour for scalar nonlinear waves

In the long-wave regime, nonlinear waves may undergo a phase transition from a smooth to a fast oscillatory behaviour. We study this phenomenon, commonly known as dispersive shock, in the light of Dubrovin's universality conjecture , and we argue that the transition can be described by a special solution of a model universal partial differential equation. This universal solution is constructed by means of a string equation. We provide a classification of universality classes and the explicit description of the transition by means of special functions, extending Dubrovin's universality conjecture to a wider class of equations. In particular, we show that Benjamin-Ono equation belongs to a novel universality class with respect to the ones known in the literature, and we compute its string equation exactly. We describe our results using the language of statistical mechanics, showing that dispersive shocks share many features of the tri-critical point in statistical systems, and building a dictionary between nonlinear waves and statistical mechanics.

hep-th↗

arXiv2012

A deformation of the method of characteristics and the Cauchy problem for Hamiltonian PDEs in the small dispersion limit

We introduce a deformation of the method of characteristics valid for Hamiltonian perturbations of a scalar conservation law in the small dispersion limit. Our method of analysis is based on the 'variational string equation', a functional-differential relation originally introduced by Dubrovin in a particular case, of which we lay the mathematical foundation. Starting from first principles, we construct the string equation explicitly up to the fourth order in perturbation theory, and we show that the solution to the Cauchy problem of the Hamiltonian PDE satisfies the appropriate string equation in the small dispersion limit. We apply our construction to explicitly compute the first two perturbative corrections of the solution to the general Hamiltonian PDE. In the KdV case, we prove the existence of a quasi-triviality transformation at any order and for arbitrary initial data.

math-ph↗