数学物理

We generalize $N \leftrightarrow -N$ duality of dimension formulae of $SU(N)$ representations on a (class of) representations with $N$-dependent Young diagrams (which include the adjoint representation), and on eigenvalues of the Casimir…

We consider series over Young diagrams of products of Schur functions $s_{\lambda\cup\lambda}$, marked with ``fat partitions'' $\lambda\cup\lambda$, which appear in matrix models associated with ensembles of symplectic and orthogonal…

We propose a generalization of the Witten conjecture, which connects a descendent enumerative theory with a specific reduction of KP integrable hierarchy. Our conjecture is realized by two parts: Part I (Geometry) establishes a…

This paper presents a novel and systematic formalism for deriving classical field equations within the framework ofcausal fermion systems, explicitly accounting for higher-order corrections such as quantum effects and those arising from…

In this paper, we are concerned with the following noncommutative Painlev\'{e} II equation \begin{equation*} \mathbf{D}^2 \beta_1 = 4\mathbf{s} \beta_1 +4 \beta_1 \mathbf{s} +8 \beta_1^3, \end{equation*} where $\beta_1=\beta_1(\vec{s})$ is…

In this paper we consider three-dimensional Schr\"odinger operators with a simple threshold eigenvalue. We show, under certain assumptions, that when a small magnetic field is introduced, this eigenvalue turns into a resonance in the…

We present a general, constructive procedure to find the basis for tensors of arbitrary order subject to linear constraints by transforming the problem to that of finding the nullspace of a linear operator. The proposed method utilizes…

In this paper, we develop the groundwork for a graph theoretic toy model of supersymmetric quantum mechanics. Using discrete Witten-Morse theory, we demonstrate that finite graphs have a natural supersymmetric structure and use this…

This work presents a geometric formulation for transforming nonconservative mechanical Hamiltonian systems and introduces a new method for regularizing and linearizing central force dynamics -- in particular, Kepler and Manev dynamics --…

We study the continuous model of the localized wave propagation corresponding to the one-dimensional diatomic crystal lattice. From the mathematical point of view the problem can be described in terms of the Cauchy problem with localized…

We study the direct and inverse scattering problems for the Zakharov-Shabat system. Representations for the Jost solutions are obtained in the form of the power series in terms of a transformed spectral parameter. In terms of that…

A fourth-order Schr\"{o}dinger equation for the description of charge transport in semiconductors in the ballistic regime is proposed with the inclusion of non-parabolic effects in the dispersion relation in order to go beyond the simple…

In this paper, we study the differential smoothness of diffusion algebras.

We present two involutivity theorems in the context of Poisson quasi-Nijenhuis %(PqN) manifolds. The second one stems from recursion relations that generalize the so called Lenard-Magri relations on a bi-Hamiltonian manifold. We apply these…

The standard Eliashberg theory of superconductivity is studied, in which the effective electron-electron interactions are mediated by generally dispersive phonons, with Eliashberg spectral function $\alpha^2 F(\omega)\geq 0$ that is…

We investigate the spectral properties of the Steklov problem for the modified Helmholtz equation $(p-\Delta) u = 0$ in the exterior of a compact set, for which the positive parameter $p$ ensures exponential decay of the Steklov…

We study the spectral properties of the Dirichlet-to-Neumann operator and the related Steklov problem in spheroidal domains ranging from a needle to a disk. An explicit matrix representation of this operator for both interior and exterior…

We consider the Bloch-Torrey operator, $-\Delta + igx$, that governs the time evolution of the transverse magnetization in diffusion magnetic resonance imaging (dMRI). Using the matrix formalism, we compute numerically the eigenvalues and…

We provide a mathematically rigorous classification of symmetry-protected topological (SPT) phases of neutral free fermions. Our approach utilizes Karoubi triples with negative squares, thought of as polarizations. We prove that neutral…

In this work, we examine the relationship between geometry and spectrum of regions with fractal boundary. The relationship is well-understood for fractal harps in one dimension, but largely open for fractal drums in larger dimensions. To…