Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0710.1089

Eigenvalue estimates for the Aharonov-Bohm operator in a domain

Rupert L. Frank·Anders Hansson
View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove semi-classical estimates on moments of eigenvalues of the Aharonov-Bohm operator in bounded two-dimensional domains. Moreover, we present a counterexample to the generalized diamagnetic inequality which was proposed by Erdos, Loss and Vougalter. Numerical studies complement these results.

Explore related subjects

math-phmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Rupert L. Frank, Anders Hansson. 2007-10-04. Eigenvalue estimates for the Aharonov-Bohm operator in a domain. https://arxiv.org/abs/0710.1089

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Thermal boundary conditions in fractional superdiffusion of energy

We study heat conduction in a one-dimensional {finite}, unpinned chain of atoms perturbed by stochastic momentum exchange and coupled to Langevin heat baths at {possibly} distinct temperatures placed at the endpoints of the chain. While infinite systems without boundaries are known to exhibit superdiffusive energy transport described by a fractional heat equation with the generator $-|Δ|^{3/4}$, the corresponding boundary conditions induced by heat baths remain less understood. We establish the hydrodynamic limit for a finite chain with $n+1$ atoms connected to thermostats at the endpoints, deriving the macroscopic evolution of the averaged energy profile. The limiting equation is governed by a non-local Lévy-type operator, with boundary terms determined by explicit interaction kernels that encode absorption, reflection, and transmission of long-wavelength phonons at the baths. Our results provide the first rigorous identification of boundary conditions for fractional superdiffusion arising directly from microscopic dynamics with local interactions, highlighting their distinction from both diffusive and pinned-chain settings

math-ph↗
arXiv2026

Two-Species Boltzmann--Kac Kinetics with Matsumoto--Yor Nonlinear Collisions: Conserved Energy and Conic Equilibria

We extend Mark Kac's kinetic framework from single-species Euclidean spaces to a two-species, non-linear interaction space on the open cone $(0, \infty)^2$ governed by the Matsumoto--Yor rational collision involution. We identify a non-linear conserved energy $E(x,y) = x + y + 1/x$ and an invariant conic metric measure $dμ= \frac{dx \, dy}{x(x+y)}$. By establishing a two-species $H$-theorem, we prove that entropy production vanishes if and only if post-collision velocities are statistically independent ($I(U;V)=0$). In contrast to the transform-domain proof of Letac and Wesołowski~\cite{letac2000}, which relies on joint Laplace transforms of general measures, our approach yields a direct physical derivation in state space via the transformation law for logarithmic gradients and separation of variables. This uncouples the functional-differential system to uniquely characterize the Generalized Inverse Gaussian (GIG) and Gamma distributions as the non-equilibrium thermodynamic equilibria.

math-ph↗