Tunnel effect in a shrinking shell enlacing a magnetic field

Abstract

Let C be a smooth planar curve. We assume that C is simple, closed, smooth, symmetric with respect to an axis and its curvature attains its minimum at exactly two points away from the axis of symmetry. In a tubular neighborhood about C, we study the Laplace operator with a magnetic flux and mixed boundary conditions. As the thickness of the domain tends to 0, we establish an explicit asymptotic formula for the splitting of the first two eigenvalues (tunneling effect).

Purely magnetic tunneling effect in two dimensions

“…The investigation will reveal the microlocal nature of the tunelling estimate given in Theorem 1.3. It contrasts with the electric tunneling à la Helffer-Sjöstrand, and even with recent contributions about purely geometric tunneling [10] and [19] where microlocal analysis is absent.…”

Tunneling effect induced by a curved magnetic edge

“…It is important to note that it is induced purely by the magnetic field, thereby providing an example of a purely magnetic quantum tunneling -where the case of [7] also required the interaction with the boundary. If we look at earlier results on the tunneling effect, with or without magnetic field, we observe that the tunneling is induced by an external potential [24,14] or by confinement to a bounded/thin domain [20,31,9]. For the Neumann realization of P h , the presence of the magnetic field adds a challenging difficulty in the estimate of the magnitude of the tunneling which was recently solved in [7].…”

“…This procedure will give us two new operators, the "right well" and "left well" operators, N ,r,γ 0 and N , ,γ 0 respectively. The same procedure appears, for similar problems in the context of geometrically induced tunneling effects [20,31], but we follow here [7, Sec. 2.4] which is slightly different, but more convenient for dealing with the symbol of the operator later on.…”

“…The importance of this step is that it allows one to rigorously approximate the bound states by the formal WKB expansions, which eventually paves the way for the analysis of an interaction matrix whose eigenvalues measure the tunneling effect. The same approach has been successfully applied in the context of thin domains [31] and the Robin Laplacian [19,20], where the proof of the tangential estimates was less technical.…”

Purely magnetic tunneling effect in two dimensions

“…We have now all the elements in hand to prove Theorem 1.3. We will follow the presentation developped in [10] to analyze the spectrum of the Robin Laplacian and be inspired by the flux considerations in [19,Section 5].…”