ar X iv : m at h - ph / 0 30 30 33 v 1 1 3 M ar 2 00 3 Strong - coupling asymptotic expansion for Schrödinger operators with a singular
نویسنده
چکیده
We investigate a class of generalized Schrödinger operators in L 2 (R 3) with a singular interaction supported by a smooth curve Γ. We find a strong-coupling asymptotic expansion of the discrete spectrum in case when Γ is a loop or an infinite bent curve which is asymptotically straight. It is given in terms of an auxiliary one-dimensional Schrödinger operator with a potential determined by the curvature of Γ. In the same way we obtain an asymptotics of spectral bands for a periodic curve. In particular, the spectrum is shown to have open gaps in this case if Γ is not a straight line and the singular interaction is strong enough.
منابع مشابه
ar X iv : m at h - ph / 0 30 30 72 v 1 3 1 M ar 2 00 3 Eigenvalue asymptotics for the Schrödinger operator with a δ - interaction on a punctured surface
Given n ≥ 2, we put r = min{ i ∈ N; i > n/2 }. Let Σ be a compact, Cr-smooth surface in Rn which contains the origin. Let further {Sǫ}0≤ǫ<η be a family of measurable subsets of Σ such that supx∈Sǫ |x| = O(ǫ) as ǫ → 0. We derive an asymptotic expansion for the discrete spectrum of the Schrödinger operator −∆−βδ(·−Σ\Sǫ) in L2(Rn), where β is a positive constant, as ǫ → 0. An analogous result is g...
متن کاملar X iv : m at h - ph / 0 30 30 45 v 1 1 9 M ar 2 00 3 p – Adic pseudodifferential operators and p – adic wavelets
We introduce a new wide class of p–adic pseudodifferential operators. We show that the basis of p–adic wavelets is the basis of eigenvectors for the introduced operators.
متن کاملar X iv : m at h - ph / 0 30 30 07 v 1 3 M ar 2 00 3 Scattering by a toroidal coil
In this paper we consider the Schrödinger operator in R 3 with a long-range magnetic potential associated to a magnetic field supported inside a torus T. Using the scheme of smooth perturbations we construct stationary modified wave operators and the corresponding scattering matrix S(λ). We prove that the essential spectrum of S(λ) is an interval of the unit circle depending only on the magneti...
متن کاملar X iv : m at h - ph / 0 20 70 25 v 1 1 9 Ju l 2 00 2 Bound states due to a strong δ interaction supported by a curved surface
We study the Schrödinger operator −∆ − αδ(x − Γ) in L 2 (R 3) with a δ interaction supported by an infinite non-planar surface Γ which is smooth, admits a global normal parameterization, and is asymptotically planar in a suitable sense. We show that for a large enough α > 0 this operator has a non-empty discrete spectrum and derive an asymptotic expansion of the eigenvalues in terms of a " two-...
متن کاملar X iv : 0 81 2 . 50 38 v 1 [ m at h . SP ] 3 0 D ec 2 00 8 SEMICLASSICAL ANALYSIS OF SCHRÖDINGER OPERATORS WITH MAGNETIC WELLS
We give a survey of some results, mainly obtained by the authors and their collaborators, on spectral properties of the magnetic Schrödinger operators in the semiclassical limit. We focus our discussion on asymptotic behavior of the individual eigenvalues for operators on closed manifolds and existence of gaps in intervals close to the bottom of the spectrum of periodic operators.
متن کامل