arXiv research

Paper introduces magnetic Hodge Laplacian for differential forms.

Researchers compute trace formula for magnetic Laplacian on hyperbolic surfaces.

Analyzes magnetic Laplacian on hyperbolic surfaces, highlighting key quantum phenomena.

In this paper, we present a Lichnerowicz type estimate and (higher order) Buser type estimates for the magnetic Laplacian on a closed Riemannian manifold with a magnetic potential. These results relate eigenvalues, magnetic fields, Ricci curvature, and Cheeger type constants.

Study magnetic Laplacians on hyperbolic surfaces, revealing three regimes of eigenfunction behavior.

Improved bounds for eigenfunctions on hyperbolic surfaces found.

Study magnetic Laplacian eigenvalues on contact manifolds.

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…

Paper derives trace formula for magnetic Laplacian at zero energy.

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)…

Novel GNN for signed and directed networks using magnetic signed Laplacian.

The paper develops a method to sparsify magnetic Laplacians using multi-type spanning forests.

Study gaps and clusters in eigenvalues of magnetic Laplacian on manifolds.

The study finds lower bounds for the first eigenvalue of the Laplacian in planar domains with magnetic fields.

On a compact Riemannian manifold $M$ with boundary, we give an estimate for the eigenvalues $(λ\_k(τ,α))\_k$ of the magnetic Laplacian with the Robin boundary conditions. Here, $τ$ is a positive number that defines the Robin condition and $α$ is a real differential 1-form on $M$ that represents the magnetic field. We e…

Upper bounds for magnetic Laplacian eigenvalues on planar domains.

We consider a Riemannian cylinder endowed with a closed potential 1-form A and study the magnetic Laplacian with magnetic Neumann boundary conditions associated with those data. We establish a sharp lower bound for the first eigenvalue and show that the equality characterizes the situation where the metric is a product…

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…

The paper identifies magnetic ground states and their role in determining the conformal class of a surface.

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

The Dirichlet Laplacian between two parallel hypersurfaces in Euclidean spaces of any dimension in the presence of a magnetic field is considered in the limit when the distance between the hypersurfaces tends to zero. We show that the Laplacian converges in a norm-resolvent sense to a Schroedinger operator on the limit…

A quasiclassical method approximates magnetic monopole eigenvalues.

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 consider a compact Riemannian manifold M endowed with a potential 1-form A and study the magnetic Laplacian associated with those data (with Neumann magnetic boundary condition if the bpoundary of M is not empty). We first establish a family of upper bounds for all the eigenvalues, compatible with the Weyl law. When…

New proofs of Cheeger-like inequalities for coexact 1-forms on hyperbolic manifolds.

We establish a uniform comparison between the spectrum of the rough Laplacian (acting on sections of a vector bundle of complex rank one or of harmonic curvature) with the spectrum of a discrete operator (a generalization of a discrete magnetic Laplacian added with a potential) acting on a finite dimensional space comi…

Study magnetic Steklov eigenvalues on manifolds with boundary.

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…

Study magnetic potentials on Anosov manifolds using spectral data.

Defines spectral varieties for non-simply connected manifolds and constructs conformal invariants.

Study sharp upper bounds for Aharonov-Bohm eigenvalues on surfaces.

We consider a connection $\nabla^X$ on a complex line bundle over a Riemann surface with boundary $M_0$, with connection 1-form $X$. We show that the Cauchy data space of the connection Laplacian (also called magnetic Laplacian) $L:={\nabla^X}^*\nabla^X + q$, with $q$ a complex valued potential, uniquely determines the…

The geometry of the quaternionic anti-de Sitter fibration is studied in details. As a consequence, we obtain formulas for the horizontal Laplacian and subelliptic heat kernel of the fibration. The heat kernel formula is explicit enough to derive small time asymptotics. Related twistor spaces and corresponding heat kern…

New method for directed graphs using learnable spectral positional encodings.

The paper calculates dimensions of higher Landau levels on compact manifolds.

We consider a compact Riemannian manifold with a Hermitian line bundle whose curvature is non-degenerate. The Laplacian acting on high tensor powers (the semiclassical regime) of the bundle exhibits a cluster of low-energy states. We demonstrate that the orthogonal projectors onto these states are the Fourier component…

This work presents a classification of all smooth 't Hooft-Jackiw-Nohl-Rebbi instantons over Gibbons-Hawking spaces. That is, we find all smooth SU(2) Yang-Mills instantons over these spaces which arise by conformal rescalings of the metric with suitable functions. Since the Gibbons-Hawking spaces are hyper-Kahler grav…

We construct pairs of compact Kähler-Einstein manifolds $(M_i,g_i,ω_i)$ ($i=1,2)$ of complex dimension $n$ with the following properties: The canonical line bundle $L_i=\bigwedge^n T^*M_i$ has Chern class $[ω_i/2π]$, and for each integer $k$ the tensor powers $L_1^{\otimes k}$ and $L_2^{\otimes k}$ are isospectral for …

Challenge to separate Earth's magnetic field from vehicle's magnetic field for accurate navigation.

Extends E. Hopf's theorem to magnetic systems without conjugate points.

A new method for spectral positional encodings in directed graphs using Hermitian block Krylov subspaces.

A magnetic field is defined by the property that its divergence is zero in a three dimensional oriented Riemannian manifold. Each magnetic field generates a magnetic flow whose trajectories are curves called as magnetic curves. In this paper, we give a new variational approach to studies the magnetic flow asociated wit…

Compatibility equations adapted to magnetic geometry.

This paper studies Hamilton-Jacobi equations for magnetic systems with constraints.

Magnetic Brunn-Minkowski inequalities on Riemannian manifolds

Study shows finiteness of magnetic hypersurfaces on closed manifolds.

The paper studies magnetic curves in $C$-manifolds and their properties.

New tensors help solve magnetic flow integrability.