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Register a new userGap control by singular Schrodinger operators in a periodically structured metamaterial
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Added by
Andrii Khrabustovskyi
Publication date
2018
fields
Physics
and research's language is
English
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We consider a family ${mathcal{H}^varepsilon}_{varepsilon>0}$ of $varepsilonmathbb{Z}^n$-periodic Schrodinger operators with $delta$-interactions supported on a lattice of closed compact surfaces; within a minimal period cell one has $minmathbb{N}$ surfaces. We show that in the limit when $varepsilonto 0$ and the interactions strengths are appropriately scaled, $mathcal{H}^varepsilon$ has at most $m$ gaps within finite intervals, and moreover, the limiting behavior of the first $m$ gaps can be completely controlled through a suitable choice of those surfaces and of the interactions strengths.
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Let $Sigmasubsetmathbb{R}^d$ be a $C^infty$-smooth closed compact hypersurface, which splits the Euclidean space $mathbb{R}^d$ into two domains $Omega_pm$. In this note self-adjoint Schrodinger operators with $delta$ and $delta$-interactions supported on $Sigma$ are studied. For large enough $minmathbb{N}$ the difference of $m$th powers of resolvents of such a Schrodinger operator and the free Laplacian is known to belong to the trace class. We prove trace formulae, in which the trace of the resolvent power difference in $L^2(mathbb{R}^d)$ is written in terms of Neumann-to-Dirichlet maps on the boundary space $L^2(Sigma)$.
Let $H_0 = -Delta + V_0(x)$ be a Schroedinger operator on $L_2(mathbb{R}^ u),$ $ u=1,2,$ or 3, where $V_0(x)$ is a bounded measurable real-valued function on $mathbb{R}^ u.$ Let $V$ be an operator of multiplication by a bounded integrable real-valued function $V(x)$ and put $H_r = H_0+rV$ for real $r.$ We show that the associated spectral shift function (SSF) $xi$ admits a natural decomposition into the sum of absolutely continuous $xi^{(a)}$ and singular $xi^{(s)}$ SSFs. This is a special case of an analogous result for resolvent comparable pairs of self-adjoint operators, which generalises the known case of a trace class perturbation while also simplifying its proof. We present two proofs -- one short and one long -- which we consider to have value of their own. The long proof along the way reframes some classical results from the perturbation theory of self-adjoint operators, including the existence and completeness of the wave operators and the Birman-Krein formula relating the scattering matrix and the SSF. The two proofs demonstrate the equality of the singular SSF with two a priori different but intrinsically integer-valued functions: the total resonance index and the singular $mu$-invariant.
We show that for a one-dimensional Schrodinger operator with a potential whose (j+1)th moment is integrable the jth derivative of the scattering matrix is in the Wiener algebra of functions with integrable Fourier transforms. We use this result to improve the known dispersive estimates with integrable time decay for the one-dimensional Schrodinger equation in the resonant case.
In this article we consider asymptotics for the spectral function of Schrodinger operators on the real line. Let $P:L^2(mathbb{R})to L^2(mathbb{R})$ have the form $$ P:=-tfrac{d^2}{dx^2}+W, $$ where $W$ is a self-adjoint first order differential operator with certain modified almost periodic structure. We show that the kernel of the spectral projector, $mathbb{1}_{(-infty,lambda^2]}(P)$ has a full asymptotic expansion in powers of $lambda$. In particular, our class of potentials $W$ is stable under perturbation by formally self-adjoint first order differential operators with smooth, compactly supported coefficients. Moreover, it includes certain potentials with dense pure point spectrum. The proof combines the gauge transform methods of Parnovski-Shterenberg and Sobolev with Melroses scattering calculus.
We study the one-dimensional Schrodinger operators $$ S(q)u:=-u+q(x)u,quad uin mathrm{Dom}left(S(q)right), $$ with $1$-periodic real-valued singular potentials $q(x)in H_{operatorname{per}}^{-1}(mathbb{R},mathbb{R})$ on the Hilbert space $L_{2}left(mathbb{R}right)$. We show equivalence of five basic definitions of the operators $S(q)$ and prove that they are self-adjoint. A new proof of continuity of the spectrum of the operators $S(q)$ is found. Endpoints of spectrum gaps are precisely described.
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