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Register a new userA geometric approximation of $delta$-interactions by Neumann Laplacians
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Added by
Andrii Khrabustovskyi
Publication date
2021
fields
Physics
and research's language is
English
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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.
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We consider a waveguide-like domain consisting of two thin straight tubular domains connected through a tiny window. The perpendicular size of this waveguide is of order $varepsilon$. Under the assumption that the window is appropriately scaled we prove that the Neumann Laplacian on this domain converges in (a kind of) norm resolvent sense as $varepsilonto 0$ to a one-dimensional Schrodinger operator corresponding to a $delta$-interaction of a non-negative strength. We estimate the rate of this convergence, also we prove the convergence of spectra.
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.
Let $Omega_-$ and $Omega_+$ be two bounded smooth domains in $mathbb{R}^n$, $nge 2$, separated by a hypersurface $Sigma$. For $mu>0$, consider the function $h_mu=1_{Omega_-}-mu 1_{Omega_+}$. We discuss self-adjoint realizations of the operator $L_{mu}=- ablacdot h_mu abla$ in $L^2(Omega_-cupOmega_+)$ with the Dirichlet condition at the exterior boundary. We show that $L_mu$ is always essentially self-adjoint on the natural domain (corresponding to transmission-type boundary conditions at the interface $Sigma$) and study some properties of its unique self-adjoint extension $mathcal{L}_mu:=overline{L_mu}$. If $mu e 1$, then $mathcal{L}_mu$ simply coincides with $L_mu$ and has compact resolvent. If $n=2$, then $mathcal{L}_1$ has a non-empty essential spectrum, $sigma_mathrm{ess}(mathcal{L}_{1})={0}$. If $nge 3$, the spectral properties of $mathcal{L}_1$ depend on the geometry of $Sigma$. In particular, it has compact resolvent if $Sigma$ is the union of disjoint strictly convex hypersurfaces, but can have a non-empty essential spectrum if a part of $Sigma$ is flat. Our construction features the method of boundary triplets, and the problem is reduced to finding the self-adjoint extensions of a pseudodifferential operator on $Sigma$. We discuss some links between the resulting self-adjoint operator $mathcal{L}_mu$ and some effects observed in negative-index materials.
We prove various estimates for the first eigenvalue of the magnetic Dirichlet Laplacian on a bounded domain in two dimensions. When the magnetic field is constant, we give lower and upper bounds in terms of geometric quantities of the domain. We furthermore prove a lower bound for the first magnetic Neumann eigenvalue in the case of constant field.
We study directed, weighted graphs $G=(V,E)$ and consider the (not necessarily symmetric) averaging operator $$ (mathcal{L}u)(i) = -sum_{j sim_{} i}{p_{ij} (u(j) - u(i))},$$ where $p_{ij}$ are normalized edge weights. Given a vertex $i in V$, we define the diffusion distance to a set $B subset V$ as the smallest number of steps $d_{B}(i) in mathbb{N}$ required for half of all random walks started in $i$ and moving randomly with respect to the weights $p_{ij}$ to visit $B$ within $d_{B}(i)$ steps. Our main result is that the eigenfunctions interact nicely with this notion of distance. In particular, if $u$ satisfies $mathcal{L}u = lambda u$ on $V$ and $$ B = left{ i in V: - varepsilon leq u(i) leq varepsilon right} eq emptyset,$$ then, for all $i in V$, $$ d_{B}(i) log{left( frac{1}{|1-lambda|} right) } geq log{left( frac{ |u(i)| }{|u|_{L^{infty}}} right)} - log{left(frac{1}{2} + varepsilonright)}.$$ $d_B(i)$ is a remarkably good approximation of $|u|$ in the sense of having very high correlation. The result implies that the classical one-dimensional spectral embedding preserves particular aspects of geometry in the presence of clustered data. We also give a continuous variant of the result which has a connection to the hot spots conjecture.
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