Eigenvalue Asymptotics Under a Non-dissipative Eigenvalue Dependent Boundary Condition for Second-order Elliptic Operators

Abstract

The asymptotic behavior of the eigenvalue sequence of the eigenvalue problem

$$ - \Delta \phi + q\left( x \right)\phi = \lambda \phi $$

in a bounded Lipschitz domain D ⊂ ℝ^N under the eigenvalue dependent boundary condition

$$ \varphi n = \sigma \lambda \varphi $$

with a continuous function Σ is investigated in the case Σ ⁻ ≢ 0, the dissipative one Σ ≥ 0 having been settled in [6]. For N = 1 the eigenvalues grow like k ² with leading asymptotic coefficient equal to the Weyl constant. For N ≥ 2 the positive eigenvalues grow like k ^2/N , while the negative eigenvalues grow in absolute value like |k|^1/(N−1). Moreover, asymptotic bounds in dependence on the dynamical coefficient function Σ are derived, firstly in the constant case, secondly for Σ of constant sign, and finally for a function Σ changing sign.