Lower bound on the spectrum of the Schrödinger operator in the plane with δ-potential supported by a curve
نویسندگان
چکیده
We consider the Schrödinger operator in the plane with δ-potential supported by a curve Γ. For the cases of an infinite curve and a finite loop we give estimates on the lower bound of the spectrum expressed explicitly through the strength of the interaction and a parameter c(Γ) which characterizes geometry of the curve Γ. Going further we cut the curve into finite number of pieces Γi and estimate the bottom of the spectrum using parameters c(Γi). As an application of the elaborated theory we consider a curve with a finite number of cusps and the general ”leaky” quantum graph.
منابع مشابه
Two Equivalent Presentations for the Norm of Weighted Spaces of Holomorphic Functions on the Upper Half-plane
Introduction In this paper, we intend to show that without any certain growth condition on the weight function, we always able to present a weighted sup-norm on the upper half plane in terms of weighted sup-norm on the unit disc and supremum of holomorphic functions on the certain lines in the upper half plane. Material and methods We use a certain transform between the unit dick and the uppe...
متن کاملEigenvalue 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 کاملNumerical Modeling and Experimental Study of Probe-Fed Rectangular Dielectric Resonator Antenna (RDRA) Supported by Finite Circular Ground Plane
Dielectric Resonator Antennas (DRAs) have received increased interest in recent years for their potential applications in microwave and millimeter wave communication systems. DRAs are normally used with the support of a ground plane. The radiation and impedance properties therefore depend not only on their physical dimensions and dielectric properties, but also on the size of the ground plane. ...
متن کاملمحاسبه خصوصیات اشباع ماده هسته ای متقارن بادرنظرگرفتن ایزوبار Δ
One of the most interesting application of the many-body methods to the nuclear physics is the calculation of the properties of the nuclear matter, especially its binding energy. In this paper, we have studied the influence of Δ -isobar on the properties of the symmetrical nuclear matter using the lowest order constrained variational (LOCV) method with the V28 potential. It is shown that the ...
متن کاملAsymptotics of the Bound State Induced by Δ-interaction Supported on a Weakly Deformed Plane
In this paper we consider the three-dimensional Schrödinger operator with a δ-interaction of strength α > 0 supported on an unbounded surface parametrized by the mapping R2 3 x 7→ (x, βf(x)), where β ∈ [0,∞) and f : R2 → R, f 6≡ 0, is a C2-smooth, compactly supported function. The surface supporting the interaction can be viewed as a local deformation of the plane. It is known that the essentia...
متن کامل