数学物理
Many novel and unique physical phenomena of incommensurate systems can be illustrated and predicted using the spectra of the associated Schr\"odinger operators. However, the absence of periodicity in these systems poses significant…
This paper concerns the derivation and validity of macroscopic descriptions of wave packets supported in the vicinity of degenerate points $(K,E)$ in the dispersion relation of tight-binding models accounting for macroscopic variations. We…
We establish the existence of spinning $Q$-vortex solitons in a complex scalar field theory with a sextic potential on a finite domain. By reducing the governing equation to a nonlinear boundary value problem, we use variational methods to…
Timelike Liouville field theory (also known as imaginary Liouville theory or imaginary Gaussian multiplicative chaos) is expected to describe two-dimensional quantum gravity in a positive-curvature regime, but its path integral is not a…
Kolmogorov-Arnold Networks (KANs) require significantly smaller architectures compared to multilayer perceptron (MLP)-based approaches, while retaining expressive power through spline-based activations. Moving boundary problems are…
We present direct and elementary commutator techniques for the Dirac equation with long-range electric and mass perturbations. The main results are absence of generalized eigenfunctions and locally uniform resolvent estimates, both in terms…
We prove that wave operators of scattering theory for fourth order Schr\"odinger operators $H = \Delta^2 + V (x)$ on $\mathbb{R}^2$ with real potentials $V(x)$ such that $\langle x \rangle^3 V(x) \in L^{\frac43}(\mathbb{R}^2)$ and $\langle…
We describe deformations of the classical principle chiral model and 1+1 Gaudin model related to ${\rm GL}_N$ Lie group. The deformations are generated by $R$-matrices satisfying the associative Yang-Baxter equation. Using the coefficients…
A pair $(K,K')$ consisting of a smooth triangulation $K$ of a compact smooth oriented Riemannian manifold $M$ and a sufficiently fine subdivision $K'$ determines a finite-dimensional Cheeger--Simons model $\mathscr{CS}(K,K')$ built from…
The theory of the parametric set-theoretic Yang-Baxter equation is established from a purely algebraic point of view. The first step towards this objective is the introduction of certain generalizations of the familiar shelves and racks…
We study the time evolution of Bose--Einstein condensates with three-body interactions in the Gross--Pitaevskii regime. We show that Bose--Einstein condensation is preserved under many-body evolution and that the condensate wavefunction…
The polyconvexity of a strain-energy function is nowadays increasingly presented as the ultimate material stability condition for an idealized elastic response. While the mathematical merits of polyconvexity are clearly understood, its…
2-Chern-Simons theory, or more commonly known as 4d BF-BB theory with gauged shift symmetry, is a natural generalization of Chern-Simons theory to 4-dimensional manifolds. It is part of the bestiary of higher-homotopy Maurer-Cartan…
Alternating the signs of the frequencies for Fourier components with even and odd (normalized) wavenumbers in the nonlinear-wave, such as the Korteweg-de Vries, equations maintains a constantly drifting dispersive shock with similar…
Some well-known examples of constrained quantum systems commonly quantized via Feynman path integrals are re-examined using the notion of conditional integrators introduced in [1]. The examples yield some new perspectives on old results. As…
We prove that the Schr\"odinger operator describing four particles in two dimensions, interacting solely through short-range three-body forces, can possess infinitely many bound states. This holds under the assumption that each three-body…
We propose a method for computing universal (in Vogel's sense) quantum dimension formulae for universal multiplets whose associated $sl$, $so$, and $sp$ representations are nonzero. The method uses the relation between $sl$ and $so$…
We study two extended Bose-Hubbard-type Hamiltonians representing bosonic networks restricted to the graph of a cube. For both Hamiltonians, we demonstrate that Bethe ansatz methods of solution can be employed after applying a canonical…
In this paper we formulate a geometric theory of elasticity and anelasticity for bodies containing material surfaces with their own elastic energies and distributed surface eigenstrains. Bulk elasticity is written in the language of…
We consider the Bogoliubov approximation for the many-body quantum dynamics of weakly interacting Bose gases and establish a uniform-in-time validity of the Bogoliubov theory. The proof relies on a detailed analysis of the dispersive…