Abstract
We investigate the spectral properties of self-adjoint Schrödinger operators with attractive δ-interactions of constant strength supported on conical surfaces in . It is shown that the essential spectrum is given by and that the discrete spectrum is infinite and accumulates to . Furthermore, an asymptotic estimate of these eigenvalues is obtained.
Highlights
We investigate the spectral properties of self-adjoint Schrödinger operators with attractive δ-interactions of constant strength α > 0 supported on conical surfaces in 3
The purpose of this paper is to analyse the spectrum of the three−dimensional Schrödinger operator −Δα, θ with an attractive δ−interaction of constant strength α > 0 supported on the conical surface
The proof of our main result is based on standard techniques in spectral theory of selfadjoint operators: we construct singular sequences and use Neumann bracketing in the spirit of [13] to show the assertion on the essential spectrum; for the infiniteness of the discrete spectrum we employ variational principles
Summary
The purpose of this paper is to analyse the spectrum of the three−dimensional Schrödinger operator −Δα, θ with an attractive δ−interaction of constant strength α > 0 supported on the conical surface. The proof of our main result is based on standard techniques in spectral theory of selfadjoint operators: we construct singular sequences and use Neumann bracketing in the spirit of [13] to show the assertion on the essential spectrum; for the infiniteness of the discrete spectrum we employ variational principles. By the definition of ωn,p we have ωn,p,+∣Cθ = ωn,p,−∣Cθ, where ωn,p,± are interpreted as rotationally invariant functions on Ω±. This implies that the first condition (9) holds. ∂ s2ωn,p,± + ∂ t2ωn,p,± + o (1), n → ∞
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