Complex $\Gamma$-convergence and magnetic Dirichlet Laplacian in bounded thin tubes

Abstract

The resolvent convergence of self-adjoint operators via the technique of Γ-convergence of quadratic forms is adapted to incorporate complex Hilbert spaces. As an application, we find effective operators to the Dirichlet Laplacian with magnetic potentials in very thin bounded tubular regions in space built along smooth closed curves; relatively weak regularity is asked for the potentials, and the convergence is in the norm resolvent sense as the cross sections of the tubes go uniformly to zero.

From Short-Range to Contact Interactions in the 1d Bose Gas

“…Proof Due to the fact that all form domains are equal, it suffices to show that, see [5,9], for all ψ ∈ H 1 (R N ), q(ψ) = lim We begin with the proof of (C.5). If ψ ∈ C ∞ 0 (R N ), then it is a fairly straightforward application of Lebesgue dominated convergence to show that (C.5) holds.…”

On the Neumann Laplacian in nonuniformly collapsing strips

“…Domain collapsing and dimensional reduction, including the confinement of quantum particles, have been studied in various aspects [2,3,5,6,7,8,9,10,11]. For example, considering the confinement in a determined region, a question of great interest is to find an effective operator when some directions are squeezed.…”

“…The rest of this work concerns the proofs of the results related to the recovery of effective operators for thick regions from approximations by uniformly collapsing ones, i.e., Theorems 3 and 4. By considering the Laplacian in Q ε and its respective quadratic form, in Section 2 we carry out the necessary change of variables to properly work with the form presented in (2). In Section 3 we discuss the restriction of the problem to the subspace L and prove Theorem 3.…”

From Short-Range to Contact Interactions in the 1d Bose Gas

“…Proof. Due to the fact that all form domains are equal, it suffices to show that, see [5,9], for all ψ ∈ H 1 (R N ), q(ψ) = lim ε→0 q ε (ψ) (C.5)…”