Mathematical Physics
The celebrated Hudson theorem states that the Gaussian functions in $\mathbb{R}^d$ are the only functions whose Wigner distribution is everywhere positive. Motivated by quantum information theory, D. Gross proved an analogous result on the…
The Levy Laplacian is an infinite-dimensional differential operator, which is interesting for its connection with the Yang-Mills gauge fields. The article proves the equivalence of various definitions of the Levy Laplacian on the manifold…
We study and classify the emergence of protected edge modes at the junction of one-dimensional materials. Using symmetries of Lagrangian planes in boundary symplectic spaces, we present a novel proof of the periodic table of topological…
In vector tomography (VT), the aim is to reconstruct an unknown multi-dimensional vector field using line integral data. In the case of a 2-dimensional VT, two types of line integral data are usually required. These data correspond to…
We study the Dirac Hamiltonian in dimension two with a mass term and a large momentum regularization, and show that bulk-edge correspondence fails. Despite a well defined bulk topological index --the Chern number--, the number of edge modes…
The time-dependent Hartree and Hartree-Fock equations provide effective mean-field descriptions for the dynamics of large fermionic systems and play a fundamental role in many areas of physics. In this work, we rigorously derive the…
We consider models of block-weighted random planar maps in which possibly decorated maps are decomposed canonically into blocks, each block receiving the weight $u$. These maps present a transition at some critical value $u=u_{cr}$ above…
We develop a geometric framework for Weyl quantization on pseudo-Riemannian manifolds, in which pseudodifferential operators act on sections of vector bundles equipped with arbitrary connections. We construct the associated star product and…
The study of undecidability in problems arising from physics has experienced a renewed interest, mainly in connection with quantum information problems. The goal of this review is to survey this recent development. After a historical…
For a two-dimensional canonical system $y'(t)=zJH(t)y(t)$ on some interval $(a,b)$ whose Hamiltonian $H$ is a.e. positive semi-definite and which is regular at $a$ and in the limit point case at $b$, denote by $q_H$ its Weyl coefficient. De…
The stability of long-range order against quenched disorder is a central problem in statistical mechanics. This paper develops a generalized framework extending the Ding-Zhuang method and integrated with the Pirogov-Sinai framework,…
Selecting an appropriate divergence measure is a critical aspect of machine learning, as it directly impacts model performance. Among the most widely used, we find the Kullback-Leibler (KL) divergence, originally introduced in kinetic…
We introduce the index $\mathcal{N}(\omega_1,\omega_2)$ of a pair of pure states on a unital C*-algebra, which is a generalization of the notion of the index of a pair of projections on a Hilbert space. We then show that the Hall…
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…