The Painlevé III equation and the Iwasawa decomposition.
The Picard group of the moduli of higher spin curves.
The principle of the largest terms and quantum large deviations
We give an approach to large deviation type asymptotic problems without evident probabilistic representation behind. An example provided by the mean field models of quantum statistical mechanics is considered.
The problems of the non-Archimedean analysis generated by quantum physics
The projected single-particle Dirac operator for Coulombic potentials.
The quantization conjecture revisited.
The quantum 3D superparticle.
The Quantum Birkhoff Normal Form and Spectral Asymptotics
In this talk we explain a simple treatment of the quantum Birkhoff normal form for semiclassical pseudo-differential operators with smooth coefficients. The normal form is applied to describe the discrete spectrum in a generalised non-degenerate potential well, yielding uniform estimates in the energy . This permits a detailed study of the spectrum in various asymptotic regions of the parameters , and gives improvements and new proofs for many of the results in the field. In the completely resonant...
The quantum duality principle
The “quantum duality principle” states that the quantization of a Lie bialgebra – via a quantum universal enveloping algebra (in short, QUEA) – also provides a quantization of the dual Lie bialgebra (through its associated formal Poisson group) – via a quantum formal series Hopf algebra (QFSHA) — and, conversely, a QFSHA associated to a Lie bialgebra (via its associated formal Poisson group) yields a QUEA for the dual Lie bialgebra as well; more in detail, there exist functors and , inverse to...
The quantum stability problem for time-periodic perturbations of the harmonic oscillator
The radiation condition at infinity for the high-frequency Helmholtz equation with source term: a wave packet approach
We consider the high-frequency Helmholtz equation with a given source term, and a small absorption parameter . The high-frequency (or: semi-classical) parameter is . We let and go to zero simultaneously. We assume that the zero energy is non-trapping for the underlying classical flow. We also assume that the classical trajectories starting from the origin satisfy a transversality condition, a generic assumption.Under these assumptions, we prove that the solution radiates in the outgoing...
The rational qKZ equation and shifted non-symmetric Jack polynomials.
The relation between Maxwell, Dirac, and the Seiberg-Witten equations.
The return of the quartic oscillator. The complex WKB method
The Role of Green’s Functions in Inverse Scattering at Fixed Energy
The role of locality in perturbation theory
The scattering problem for a noncommutative nonlinear Schrödinger equation.
The Schrödinger equation on non-stationary domains.
The second-order Klein-Gordon field equation.
The Semiclassical Electron in a Magnetic Field and Lattice. Some Problems of Low Dimensional "Periodic" Topology.