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In this paper we investigate the operator H β = −∆ − βδ(· − Γ) in L 2 (R 2), where β > 0 and Γ is a closed C 4 Jordan curve in R 2. We obtain the asymptotic form of each eigenvalue of H β as β tends to infinity. We also get the asymptotic form of the number of negative eigenvalues of H β in the strong coupling asymptotic regime.

- K. Yoshitomi
- 2003

Given n ≥ 2, we put r = min{ i ∈ N; i > n/2 }. Let Σ be a compact, C r-smooth surface in R n which contains the origin. Let further {S ǫ } 0≤ǫ<η be a family of measurable subsets of Σ such that sup x∈Sǫ |x| = O(ǫ) as ǫ → 0. We derive an asymptotic expansion for the discrete spectrum of the Schrödinger operator −∆ − βδ(· − Σ \ S ǫ) in L 2 (R n), where β is a… (More)

- K. Yoshitomi
- 2001

In this paper we study the operator H β = −∆ − βδ(· − Γ) in L 2 (R 2), where Γ is a smooth periodic curve in R 2. We obtain the asymptotic form of the band spectrum of H β as β tends to infinity. Furthermore, we prove the existence of the band gap of σ(H β) for sufficiently large β > 0. Finally, we also derive the spectral behaviour for β → ∞ in the case… (More)

- K. Yoshitomi
- 2002

We investigate the two-dimensional magnetic Schrödinger operator H B,β = (−i∇ − A) 2 − βδ(· − Γ), where Γ is a smooth loop and the vector potential A corresponds to a homogeneous magnetic field B perpendicular to the plane. The asymptotics of negative eigenvalues of H B,β for β → ∞ is found. It shows, in particular, that for large enough positive β the… (More)

In this paper we investigate the operator H = ??(??) in L 2 (R 2), where > 0 and ? is a closed C 4 Jordan curve in R 2. We obtain the asymptotic form of each eigenvalue of H as tends to innnity. We also get the asymptotic form of the number of negative eigenvalues of H in the strong coupling asymptotic regime.

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