arXiv research

We solved the Schr{ö}dinger equation for a particle in a uniform magnetic field in the n-dimensional torus. We obtained a complete set of solutions for a broad class of problems; the torus T^n = R^n / Λ is defined as a quotient of the Euclidean space R^n by an arbitrary n-dimensional lattice Λ. The lattice is not neces…

The study establishes bounds for Schrödinger operators on Riemannian manifolds.

Characterizes Schrödinger operator boundedness on weighted Riemannian manifolds.

Abstract notes on generative modeling techniques.

Paper introduces a new generative learning model using Schrödinger bridge diffusion in latent space.

Unified framework for robust, stable, and efficient density ratio estimation.

We give a new lower bound for the first gap $λ_2 - λ_1$ of the Dirichlet eigenvalues of the Schr{ö}dinger operator on a bounded convex domain $Ω$ in R$^n$ or S$^n$ and greatly sharpens the previous estimates. The new bound is explicit and computable.

In this paper, we partially settle down the long standing open problem of the finite time blow-up property about the nonlinear Schr$\ddot{o}$dinger equations on some Riemannian manifolds like the standard 2-sphere $S^2$ and the hyperbolic 2-space $H^{2}(-1)$. Using the similar idea, we establish such blow-up results on…

The goal of this article is twofold: in a first part, we prove Gaussian estimates for the heat kernel of Schr{ö}dinger operators delta + V whose potential V is "small at infinity" in an integral sense. In a second part, we prove sharp boundedness result for the associated Riesz transform with potential d(delta+V) --1/2…

For the spherical Laplacian on the sphere and for the Dirichlet Laplacian in the square}, Antonie Stern claimed in her PhD thesis (1924) the existence of an infinite sequence of eigenvalues whose corresponding eigenspaces contain an eigenfunction with exactly two nodal domains. These results were given complete proofs …

Study shows observability for Schrödinger equations on product manifolds with specific conditions.

Suppose that $G=(V, E)$ is a finite graph with the vertex set $V$ and the edge set $E$. Let $Δ$ be the usual graph Laplacian. Consider the following nonlinear Schr$\ddot{o}$dinger type equation of the form $\left \{ \begin{array}{lcr} -Δu-αu=f(x,u),\\ u\in W^{1,2}(V),\\ \end{array} \right.$ on graph $G$, where $f(x…

New inequalities for spectral zeta kernels on spheres and manifolds.

Generative model for time series using Schrödinger bridge.

CMCD sampler connects transport and variational inference for efficient sampling.

Study on magnetic Dirac operators and their spectrum.

The article studies eigenvalues and spectrum of magnetic Dirac operators.

Paper introduces magnetic Steklov operator on differential forms and its properties.

Study magnetic Schrödinger operators in Euclidean space.

Study magnetic Steklov eigenvalues on manifolds with boundary.

Guillarmou extends X-ray transform to magnetic and thermostat flows.

We consider a magnetic Schrödinger operator $H^h$, depending on a semiclassical parameter $h>0$, on a compact Riemannian manifold. We assume that there is no electric field. We suppose that the minimal value $b_0$ of the intensity of the magnetic field $b$ is strictly positive. We give a survey of the results on asympt…

The paper studies magnetic curvature and proves the existence of closed orbits on low energy levels.

Study magnetic potentials on Anosov manifolds using spectral data.

Constructs smooth integrable magnetic systems on a two-torus.

Study sharp lower bounds on negative eigenvalues of magnetic Pauli operator.

The Guillemin-Uribe trace formula is a semiclassical version of the Selberg trace formula and more general Duistermaat-Guillemin formula for elliptic operators on compact manifolds, which reflects the dynamics of magnetic geodesic flows in terms of eigenvalues of a natural differential operator (the magnetic Laplacian)…

A quasiclassical approximation is constructed to describe the eigenvalues of the magnetic Laplacian on a compact Riemannian manifold in the case when the magnetic field is not given by an exact 2-form. For this, the multidimensional WKB method in the form of Maslov canonical operator is applied. In this case, the canon…

We consider an open domain with a compact boundary in an Euclidean space and a Schroedinger operator with magnetic field on this domain. We give sufficient conditions on the rate of growth of the magnetic field near the boundary which guarantees essential self-adjointness of this operator. From the physical point of vi…

Paper introduces magnetic Hodge Laplacian for differential forms.

Magnetic Brunn-Minkowski inequalities on Riemannian manifolds

We investigate nodal sets of magnetic Schroedinger operators with zero magnetic field, acting on a non simply connected domain in $\r^2$. For the case of circulation 1/2 of the magnetic vector potential around each hole in the region, we obtain a charactisation of the nodal set, and use this to obtain bounds on the mul…

This study explores magnetic trajectories on the Heisenberg group, finding symmetries and solutions.

We consider open manifolds which are interiors of a compact manifold with boundary, and Riemannian metrics asymptotic to a conformally cylindrical metric near the boundary. We show that the essential spectrum of the Laplace operator on functions vanishes under the presence of a magnetic field which does not define an i…

We propose a twistor construction of surfaces in Lie sphere geometry based on the linear system which copies equations of Wilczynski's projective frame. In the particular case of Lie-applicable surfaces this linear system describes joint eigenfunctions of a pair of commuting Schrödinger operators with magnetic fields.

We consider the Bochner Laplacian on high tensor powers of a positive line bundle on a closed symplectic manifold (or, equivalently, the semiclassical magnetic Schrödinger operator with the non-degenerate magnetic field). We assume that the operator has discrete wells. The main result of the paper states asymptotic exp…

New theorem shows gaps in magnetic Schrödinger operator spectra for large coupling.

Study magnetic flows on 3D contact sub-Riemannian manifolds using Rumin complex.

We consider the semi-classical Dirac operator coupled to a magnetic potential on a large class of manifolds including all metric contact manifolds. We prove a sharp local Weyl law and a bound on its eta invariant. In the absence of a Fourier integral parametrix, the method relies on the use of almost analytic continuat…

Study of gyroscopic Chaplygin systems and magnetic flows on spheres.

Given a constant magnetic field on Euclidean space ${\mathbb R}^p$ determined by a skew-symmetric $(p\times p)$ matrix $Θ$, and a ${\mathbb Z}^p$-invariant probability measure $μ$ on the disorder set $Σ$ which is by hypothesis a Cantor set, where the action is assumed to be minimal, the corresponding Integrated Density…

Symmetry-electronic fingerprints reveal competing magnetic phases in two-dimensional materials.

We study the eigenvalues of the magnetic Schroedinger operator associated with a magnetic potential A and a scalar potential q, on a compact Riemannian manifold M, with Neumann boundary conditions if the boundary is not empty. We obtain several bounds for the spectrum. Besides the dimension and the volume of the manifo…

We propose a new definition for the abelian magnetic charge density of a non-abelian monopole, based on zero-modes of an associated Dirac operator. Unlike the standard definition of the charge density, this density is smooth in the core of the monopole. We show that this charge density induces a magnetic field whose ex…

We study spectral properties of the Laplace-Beltrami operator on two relevant almost-Riemannian manifolds, namely the Grushin structures on the cylinder and on the sphere. This operator contains first order diverging terms caused by the divergence of the volume. We get explicit descriptions of the spectrum and the eige…

We present a novel algorithm that predicts the probability that the time derivative of the horizontal component of the ground magnetic field $dB/dt$ exceeds a specified threshold at a given location. This quantity provides important information that is physically relevant to Geomagnetically Induced Currents (GIC), whic…

A necessary and sufficient condition for energy-momentum conservation is proved within a topological, pre-metric approach to classical electrodynamics including magnetic as well as electric charges. The extended Lorentz force, consisting of mutual actions by F=(E, B) on the electric current and G=(H, D) on the magnetic…

Exponential localization of eigensections for Bochner-Schrödinger operator.