Upper bounds for the ground state energy of the Laplacian with zero magnetic field on planar domains

We obtain upper bounds for the first eigenvalue of the magnetic Laplacian associated to a closed potential 1-form (hence, with zero magnetic field) acting on complex functions of a planar domain $$\Omega $$ Ω , with magnetic Neumann boundary conditions. It is well known that the first eigenvalue is positive whenever the potential admits at least one non-integral flux. By gauge invariance, the lowest eigenvalue is simply zero if the domain is simply connected; then, we obtain an upper bound of the ground state energy depending only on the ratio between the number of holes and the area; modulo a numerical constant the upper bound is sharp and we show that in fact equality is attained (modulo a constant) for Aharonov-Bohm-type operators acting on domains punctured at a maximal $$\epsilon $$ ϵ -net. In the last part, we show that the upper bound can be refined, provided that one can transform the given domain in a simply connected one by performing a number of cuts with sufficiently small total length; we thus obtain an upper bound of the lowest eigenvalue by the ratio between the number of holes and the area, multiplied by a Cheeger-type constant, which tends to zero when the domain is metrically close to a simply connected one.

Geometric and spectral estimates based on spectral Ricci curvature assumptions

We obtain a Bonnet–Myers theorem under a spectral condition: a closed Riemannian {(M^{n},g)} manifold for which the lowest eigenvalue of the Ricci tensor ρ is such that the Schrödinger operator {\Delta+(n-2)\rho} is positive has finite fundamental group. Further, as a continuation of our earlier results, we obtain isoperimetric inequalities from Kato-type conditions on the Ricci curvature. We also obtain the Kato condition for the Ricci curvature under purely geometric assumptions.

Counterexample to Strong Diamagnetism for the Magnetic Robin Laplacian

We determine a counterexample to strong diamagnetism for the Laplace operator in the unit disc with a uniform magnetic field and Robin boundary condition. The example follows from the accurate asymptotics of the lowest eigenvalue when the Robin parameter tends to $-\infty $ − ∞ .

Complete Asymptotics for Solution of Singularly Perturbed Dynamical Systems with Single Well Potential

We consider a singularly perturbed boundary value problem ( − ε 2 ∆ + ∇ V · ∇ ) u ε = 0 in Ω , u ε = f on ∂ Ω , f ∈ C ∞ ( ∂ Ω ) . The function V is supposed to be sufficiently smooth and to have the only minimum in the domain Ω . This minimum can degenerate. The potential V has no other stationary points in Ω and its normal derivative at the boundary is non-zero. Such a problem arises in studying Brownian motion governed by overdamped Langevin dynamics in the presence of a single attracting point. It describes the distribution of the points at the boundary ∂ Ω , at which the trajectories of the Brownian particle hit the boundary for the first time. Our main result is a complete asymptotic expansion for u ε as ε → + 0 . This asymptotic is a sum of a term K ε Ψ ε and a boundary layer, where Ψ ε is the eigenfunction associated with the lowest eigenvalue of the considered problem and K ε is some constant. We provide complete asymptotic expansions for both K ε and Ψ ε ; the boundary layer is also an infinite asymptotic series power in ε . The error term in the asymptotics for u ε is estimated in various norms.

Transfer Matrix of the Two-dimensional Ising Model

This chapter deals with the exact solution of the two-dimensional Ising model as it is achieved through the transfer matrix formalism. It discusses the crucial role played by the commutative properties of the transfer matrices, which lead to a functional equation for their eigenvalues. The exact free energy of the Ising model and its critical point can be identified by means of the lowest eigenvalue. The chapter covers Baxter's approach, the Yang–Baxter equation and its relation to the Boltzmann weights, the R-matrix, and discusses activity away from the critical point, the six-vertex model, as well as functional equations and symmetries.

On the Sampson Laplacian

In the present paper we consider the little-known Sampson operator that is strongly elliptic and self-adjoint second order differential operator acting on covariant symmetric tensors on Riemannian manifolds. First of all, we review the results on this operator. Then we consider the properties of the Sampson operator acting on one-forms and symmetric two-tensors. We study this operator using the analytical method, due to Bochner, of proving vanishing theorems for the null space of a Laplace operator admitting a Weitzenb?ck decomposition. Further we estimate operator?s lowest eigenvalue.