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Register a new userDiscrete spectrum of interactions concentrated near conical surfaces
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Added by
Konstantin Pankrashkin
Publication date
2016
fields
Physics
and research's language is
English
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We study the spectrum of two kinds of operators involving a conical geometry: the Dirichlet Laplacian in conical layers and Schrodinger operators with attractive $delta$-interactions supported by infinite cones. Under the assumption that the cones have smooth cross-sections, we prove that such operators have infinitely many eigenvalues accumulating below the threshold of the essential spectrum and we express the accumulation rate in terms of the eigenvalues of an auxiliary one-dimensional operator with a curvature-induced potential.
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We discuss several geometric conditions guaranteeing the finiteness or the infiniteness of the discrete spectrum for Robin Laplacians on conical domains.
Consider a quantum particle trapped between a curved layer of constant width built over a complete, non-compact, $mathcal C^2$ smooth surface embedded in $mathbb{R}^3$. We assume that the surface is asymptotically flat in the sense that the second fundamental form vanishes at infinity, and that the surface is not totally geodesic. This geometric setting is known as a quantum layer. We consider the quantum particle to be governed by the Dirichlet Laplacian as Hamiltonian. Our work concerns the existence of bound states with energy beneath the essential spectrum, which implies the existence of discrete spectrum. We first prove that if the Gauss curvature is integrable, and the surface is weakly $kappa$-parabolic, then the discrete spectrum is non-empty. This result implies that if the total Gauss curvature is non-positive, then the discrete spectrum is non-empty. We next prove that if the Gauss curvature is non-negative, then the discrete spectrum is non-empty. Finally, we prove that if the surface is parabolic, then the discrete spectrum is non-empty if the layer is sufficiently thin.
We consider the self-adjoint Landau Hamiltonian $H_0$ in $L^2(mathbb{R}^2)$ whose spectrum consists of infinitely degenerate eigenvalues $Lambda_q$, $q in mathbb{Z}_+$, and the perturbed operator $H_upsilon = H_0 + upsilondelta_Gamma$, where $Gamma subset mathbb{R}^2$ is a regular Jordan $C^{1,1}$-curve, and $upsilon in L^p(Gamma;mathbb{R})$, $p>1$, has a constant sign. We investigate ${rm Ker}(H_upsilon -Lambda_q)$, $q in mathbb{Z}_+$, and show that generically $$0 leq {rm dim , Ker}(H_upsilon -Lambda_q) - {rm dim , Ker}(T_q(upsilon delta_Gamma)) < infty,$$ where $T_q(upsilon delta_Gamma) = p_q (upsilon delta_Gamma)p_q$, is an operator of Berezin-Toeplitz type, acting in $p_q L^2(mathbb{R}^2)$, and $p_q$ is the orthogonal projection on ${rm Ker},(H_0 -Lambda_q)$. If $upsilon eq 0$ and $q = 0$, we prove that ${rm Ker},(T_0(upsilon delta_Gamma)) = {0}$. If $q geq 1$, and $Gamma = mathcal{C}_r$ is a circle of radius $r$, we show that ${rm dim , Ker} (T_q(delta_{mathcal{C}_r})) leq q$, and the set of $r in (0,infty)$ for which ${rm dim , Ker}(T_q(delta_{mathcal{C}_r})) geq 1$, is infinite and discrete.
We demonstrate how to approximate one-dimensional Schrodinger operators with $delta$-interaction by a Neumann Laplacian on a narrow waveguide-like domain. Namely, we consider a domain consisting of a straight strip and a small protuberance with room-and-passage geometry. We show that in the limit when the perpendicular size of the strip tends to zero, and the room and the passage are appropriately scaled, the Neumann Laplacian on this domain converges in (a kind of) norm resolvent sense to the above singular Schrodinger operator. Also we prove Hausdorff convergence of the spectra. In both cases estimates on the rate of convergence are derived.
We consider the Schrodinger operator $H_{eta W} = -Delta + eta W$, self-adjoint in $L^2({mathbb R}^d)$, $d geq 1$. Here $eta$ is a non constant almost periodic function, while $W$ decays slowly and regularly at infinity. We study the asymptotic behaviour of the discrete spectrum of $H_{eta W}$ near the origin, and due to the irregular decay of $eta W$, we encounter some non semiclassical phenomena. In particular, $H_{eta W}$ has less eigenvalues than suggested by the semiclassical intuition.
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