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 with a uniformly elliptic metric. We show that if Γ is asymptotically planar in a suitable sense and α > 0 is sufficiently large this operator has a non-empty discrete spectrum and derive an asymptotic expansion of the eigenvalues in iterms of a "two-dimensional" comparison operator determined by the geometry of the surface Γ.
We study the Laplacian in L 2 (Ê 3 ) perturbed on an infinite curve Γ by a δ interaction defined through boundary conditions which relate the corresponding generalized boundary values. We show that if Γ is smooth and not a straight line but it is asymptotically straight in a suitable sense, and if the interaction does not vary along the curve, the perturbed operator has at least one isolated eigenvalue below the threshold of the essential spectrum. P. Exner and S. KondejAnn. Henri Poincaré other tools. One possibility is to employ the resolvent formula for a curve in R 3 derived in [Ku]. However, since it uses rather strong regularity hypotheses about the curve we take another route and begin instead with an abstract formula for strongly singular perturbations due to A. Posilicano [Po1]. When it is specified to our particular case, it contains again an embedding operator into a space of functions supported on the curve Γ, however, this time it is not the "naive" L 2 but rather a suitable element from the scale of Sobolev spaces. Of course, one can regard it as a generalization of Krein's formula; recall that such a way of expressing the resolvent can be used not only to describe δ interaction perturbations but also more general dynamics supported by zero measure sets [Ka, KK, Ko]. Another aspect of the absence of a description in terms of the quadratic-form sum concerns the very definition of the operator we want to study. We have to employ boundary conditions which relate the corresponding generalized boundary values in the normal plane to the curve modeled after the usual two-dimensional δ interaction [AGHH], which requires us to impose stronger regularity conditions on Γ. Furthermore, a modification of the Birman-Schwinger technique used in [EI] demands stronger restrictions on the regularity of the curve. On the other hand, apart of these technical hypotheses our main result -stated in Theorem 5.6 below -is analogous to that of [EI], namely that for any curve which is asymptotically straight but not a straight line the corresponding operator has at least one isolated eigenvalue. This conclusion is by no means obvious having in mind how different are the point interactions in one and two dimensions.which are obviously bounded too. Let Z be an open subset of (−∆) symmetric w.r.t. the real axis, i.e. such that z ∈ Z impliesz ∈ Z. Suppose that for any Vol. 3, 2002 Curvature-Induced Bound States 969 z ∈ Z there exists a closed operator Q z : D ⊆ X → X satisfying the following conditions,
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 the 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 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.1) however, a proper way to define the operator corresponding to the formal expression is involved and will be explained in Sec. 2.2. a A physical motivation for this model a In particular, this is the reason why we use here a formal coupling constant different from the parameter α introduced in the condition (2.4) below. 559 Rev. Math. Phys. 2004.16:559-582. Downloaded from www.worldscientific.com by UNIVERSITY OF SASKATCHEWAN on 02/04/15. For personal use only. 560 P. Exner & S. Kondejis to understand the electron behavior in "leaky" quantum wires, i.e. a model of these semiconductor structures which is realistic since it takes into account the fact that the electron as a quantum particle capable of tuneling can be found outside the wire (cf.[8]) for a more detailed discussion. One natural question is whether in the case of a strong transverse coupling, properties of such a "leaky" wire will approach those of an ideal wire of zero thickness, i.e. the model in which the particle is confined to Γ alone, and how the geometry of the configuration manifold will be manifested at that. In the two-dimensional case when Γ is a planar curve, this problem was analyzed in [11,12] where it was shown that apart from the divergent term which describes the energy of coupling to the curve, the spectrum coincides asymptotically with that of an auxiliary onedimensional Schrödinger operator with a curvature-induced potential. bThe case of a curve in R 3 which we are going to discuss here is more complicated for several reasons. First of all, the codimension of Γ is two in this situation which means that to define the Hamiltonian, we cannot use the natural quadratic form and have to employ generalized boundary conditions instead. Furthermore, while the strategy of [11, 12] based on bracketing bounds combined with the use of suitable curvilinear coordinates in the vicinity of Γ can be applied again, the "straightening" transformation we have to employ is more involved here. Also the bound on the transverse part of the estimating operators is less elementary in this case.Let us review briefly the contents of the paper. We begin by constructing a self-adjoint operator H α,Γ which corresponds to the formal expression (1.1), where Γ is a curve in R 3 ; this will be done in Sec. 2.5. To this aim, we employ in the transverse plane to Γ the usual boundary conditions defining a two-dimensional point interaction [...
We discuss a generalized Schrödinger operator in We analyze the discrete spectrum, and furthermore, we show that the resonance problem in this setting can be explicitly solved; by BirmanSchwinger method it is cast into a form similar to the Friedrichs model.
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