Magnetic spectral bounds on starlike plane domains

Abstract

We develop sharp upper bounds for energy levels of the magnetic Laplacian on starlike plane domains, under either Dirichlet or Neumann boundary conditions and assuming a constant magnetic field in the transverse direction. Our main result says that n j=1 Φ λ j A/G is maximal for a disk whenever Φ is concave increasing, n ≥ 1, the domain has area A, and λ j is the j-th Dirichlet eigenvalue of the magnetic Laplacian i∇+ β 2A (−x 2 , x 1 ) 2 . Here the flux β is constant, and the scale invariant factor G penalizes deviations from roundness, meaning G ≥ 1 for all domains and G = 1 for disks.

Steklov Eigenvalues and Quasiconformal Maps of Simply Connected Planar Domains

“…Motivated by this physical interpretation, and in an attempt to prove a summed version of the Pólya conjecture, the eigenvalue sums of the Laplacian have been studied extensively through Berezin-Li-Yau inequalities [19,35], giving results that are asymptotically sharp as j → ∞. In a different direction, geometrically sharp inequalities for Laplace eigenvalue sums (with fixed index j) were developed on starlike domains by the second and third authors [32,33]. The biLaplacian was treated too [39].…”

Lower bounds for the first eigenvalue of the magnetic Laplacian

“…K Ω,ψ ≤ sup Ω g 11 inf Ω g 11 . Note in particular that if Ω is rotationally invariant, so that the metric can be put in the form:…”

“…Assume that equality holds. Then, if u is an eigenfunction, we know that u = u(t) by the discussion in (11) and u restricts to an eigenfunction on each level circle Σ r for the potential A = H(t) dt above (see Fact 3 at the beginning of Section 2.3 and the second step above).…”

“…The spectrum of the magnetic Laplacian is very much studied in analysis (see for example [3] and the references therein) and in relation with physics. For Dirichlet boundary conditions, lower estimates of its fundamental tone have been worked out, in particular, when Ω is a planar domain and B is the constant magnetic field; that is, when the function B is constant on Ω (see for example a Faber-Krahn type inequality in [8] and the recent [11] and the references therein, also for Neumann boundary condition). The case when the potential A is a closed 1-form is particularly interesting from the physical point of view (Aharonov-Bohm effect), and also from the geometric point of view.…”

Generalized tight p-frames and spectral bounds for Laplace-like operators

“…See Theorem 1.9 for an application of this approach. Note also that the Haar measure averaging generalizes to nonlinear transformations of balls [38,39], assuming f (t) = t.…”