An effective Hamiltonian for the eigenvalue asymptotics of the Robin Laplacian with a large parameter

Abstract

Abstract. We consider the Laplacian on a class of smooth domains Ω ⊂ R ν , ν ≥ 2, with attractive Robin boundary conditions:where n is the outer unit normal, and study the asymptotics of its eigenvalues Ej(Q Ω α ) as well as some other spectral properties for α → +∞ We work with both compact domains and non-compact ones with a suitable behavior at infinity. For domains with compact C 2 boundaries and fixed j, we show thatwhere µj(α) is the j th eigenvalue, as soon as it exists, of −∆S−(ν−1)αH with (−∆S) and H being respectively the positive Laplace-Beltrami operator and the mean curvature on ∂Ω. Analogous results are obtained for a class of domains with non-compact boundaries.In particular, we discuss the existence of eigenvalues in non-compact domains and the existence of spectral gaps for periodic domains. We also show that the remainder estimate can be improved under stronger regularity assumptions. The effective Hamiltonian −∆S − (ν − 1)αH enters the framework of semi-classical Schrödinger operators on manifolds, and we provide the asymptotics of its eigenvalues in the limit α → +∞ under various geometrical assumptions. In particular, we describe several cases for which our asymptotics provides gaps between the eigenvalues of Q IntroductionLet Ω ⊂ R ν , ν ≥ 2, be an open set with a sufficiently regular boundary S := ∂Ω. For α ∈ R, denote by Q Ω α the operator Q Ω α u = −∆u on the functions u defined in Ω and satisfying the Robin boundary condition ∂u ∂nwhere n is the outer unit normal at S. More precisely,where dS stands for the (ν − 1)-dimensional Hausdorff measure on S, which is closed and semibounded from below under suitable assumptions (e.g. if S is compact or with a suitable behavior at infinity, see below), and we denote by E Ω j (Q Ω α ) the j th eigenvalue of Q Ω α below the bottom of the essential spectrum, as soon as it exists. The aim of the paper is to obtain new results on the asymptotics of the eigenvalues as α tends to +∞.The problem appears in various applications, such as reaction-diffusion processes [25] and the enhanced surface superconductivity [14], and the related questions were already 1 2 discussed in the previous works by various authors. Let us present briefly the state of art for compact domains. It was shown in [25,26] that for piecewise smooth Liptschotz domain one has E 1 (Q Ω α ) = −C Ω α 2 + o(α 2 ) as α → +∞, where C Ω ≥ 1 is a constant depending on the geometric properties of Ω. In particular, C Ω = 1 for C 1 domains, see [5,27]. More detailed asymptotic expansions for some specific non-smooth domains were considered in [17,26,31]. As for smooth domains, a more detailed result was obtained first in [9,30] for ν = 2 and then in [32] for any ν ≤ 2: if the domain is C 3 and j ∈ N is fixed, thenwhere H max is the maximum of the mean curvature H of the boundary (the exact definition will be recalled below). Remark that this asymptotics together with isompermetric inequalities for the mean curvature have played an important role for the so-called reverse Faber-Krahn inequality, ...

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Robin eigenvalues on domains with peaks

“…has the same order in α, which is in contrast to the previously studied cases with more regularity: as shown in [10], for curvilinear polygons one has G j = O(α 2 ), and for C k smooth domains one has [20].…”

Thermodynamic Properties of the 1D Robin Quantum Well

“…with γ from Equation (16). Of course, similar to the canonical ensemble, at = ∞, it degenerates to its Neumann counterpart.…”

“…The mathematical and physical reasons for this strong binding in general n-dimensional domain were explained and analyzed before [9][10][11][12][13][14][15][16][17][18] and repeated for our geometry in a preceding paper [8] where also quantum-information measures of the structure were computed. The mathematical and physical reasons for this strong binding in general n-dimensional domain were explained and analyzed before [9][10][11][12][13][14][15][16][17][18] and repeated for our geometry in a preceding paper [8] where also quantum-information measures of the structure were computed.…”