Sum of the negative eigenvalues for the semi-classical Robin Laplacian
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Sum of the negative eigenvalues for the semi-classical Robin Laplacian Ayman Kachmar1 · Marwa Nasrallah2 Received: 12 October 2018 / Accepted: 1 December 2019 © Universidad Complutense de Madrid 2019
Abstract We compute the sum of the negative eigenvalues of a semi-classical version of the Robin Laplace operator. This version of the operator arises naturally from the Laplace operator with a Robin boundary condition and a strong coupling parameter. Viewing the operator from the ‘semi-classical’ angle has allowed for many non-trivial results. In the same vein, this contribution is no exception with the main significance being that the function in the boundary condition satisfies minimal regularity assumptions. Earlier contributions were devoted to the case when the function in the boundary condition is constant. Keywords Laplace operator · Number of bound states · Variational principles Mathematics Subject Classification 35P15 · 35P20 · 47A75
1 Introduction 1.1 The semi-classical Robin Laplacian This paper studies the Robin Laplace operator, Lh,γ , defined starting from the following semi-bounded closed quadratic form, Qh,γ (u) :=
B
|h∇u|2 d x − h 3/2
∂
γ (x)|u(x)|2 dσ (x) ,
(1.1)
Marwa Nasrallah [email protected] Ayman Kachmar [email protected]
1
Department of Mathematics, Lebanese University, Nabatiye, Lebanon
2
Faculty of Sciences IV, Lebanese University, Zahle, Lebanon
123
A. Kachmar, M. Nasrallah
where • • • •
⊂ R2 is an open domain with a smooth C 3 and compact boundary ∂; γ ∈ L 3 (∂); h > 0 is a parameter (the semi-classical parameter); dσ (x) is the arc-length measure on ∂.
The quadratic form in (1.1) is defined on the Sobolev space H 1 () = {u ∈ L 2 () : ∇u ∈ L 2 ()}. The assumption γ ∈ L 3 () ensures that this quadratic form is semi-bounded and closed (cf. [19, Lemma A.1]), hence it generates, by the Friedrichs theorem, the self-adjoint operator Lh,γ = −h 2 defined on the following domain, Dom(Lh,γ ) = {u ∈ H 2 () : n · h 1/2 ∇u + γ u = 0 on ∂},
(1.2)
where n is the inward pointing normal to the boundary of . The spectral theory of the Robin Laplacian has potential applications in the study of superconductors [11,16] and also in reaction-diffusion systems [34]. Let us point out the following relationship between the operator Lh,γ and the Robin Laplace operator with a strong coupling parameter σ , which has attracted the attention of many authors recently; it is defined as follows, Tσ,γ = − on Dom(Tσ,γ ) = {u ∈ H 2 () : n · ∇u + σ γ u = 0 on ∂} . Taking σ = h −2 , we obtain the semi-classical operator Lh,γ along with the equivalence between the strong coupling limit σ → +∞ and the semi-classical limit h → 0+ . 1.2 Historical background Having the rich literature on the semi-classical methods in hand, one can obtain a lot of non-trivial results on the operator Lh,γ . All the earlier results where devoted to the case where γ = 1 but can be generalized to the case of a smooth non-constant boundary function γ . Starting with the contributions by Levitin–P
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