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# Gap control by singular Schr\"odinger operators in a periodically structured metamaterial

**Preprint**· February 2018

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Abstract

We consider a family $\{\mathcal{H}_\varepsilon\}_{\varepsilon}$ of $\varepsilon\mathbb{Z}^n$-periodic Schr\"odinger operators with $\delta'$-interactions supported on a lattice of closed compact surfaces; within a minimal period cell one has $m\in\mathbb{N}$ surfaces. We show that in the limit when $\varepsilon\to 0$ 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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- The main result of this work is as follows: for arbitrary pairwise disjoint finite intervals $(\alpha_j,\beta_j)\subset[0,\infty)$, $j=1,\dots,m$ and for arbitrary $n\geq 2$ we construct the family of periodic non-compact domains $\{\Omega^\varepsilon\subset\mathbb{R}^n\}_{\varepsilon>0}$ such that the spectrum of the Neumann Laplacian in $\Omega^\varepsilon$ has at least $m$ gaps when $\varepsilon$ is small enough, moreover the first $m$ gaps tend to the intervals $(\alpha_j,\beta_j)$ as $\varepsilon\to 0$. The constructed domain $\Omega^\varepsilon$ is obtained by removing from $\mathbb{R}^n$ a system of periodically distributed "trap-like" surfaces.
- Floquet-Bloch theory for elliptic problems with discontinuous coefficients In: Spectral theory and analysis, Oper. Theory Adv
- Jan 2011
- 1-20

- B M Brown
- V Hoang
- M Plum
- I G Wood

B.M. Brown, V. Hoang, M. Plum, I.G. Wood, Floquet-Bloch theory for elliptic problems with discontinuous coefficients. In: Spectral theory and analysis, Oper. Theory Adv. Appl. 214, Birkhäuser, Basel, 2011, 1-20. - ArticleFull-text available
- Jun 2015

We consider a family of quantum graphs $\{(\Gamma,\mathcal{A}_\varepsilon)\}_{\varepsilon>0}$, where $\Gamma$ is a $\mathbb{Z}^n$-periodic metric graph and the periodic Hamiltonian $\mathcal{A}_\varepsilon$ is defined by the operation $-\varepsilon^{-1} {\mathrm{d} ^2\over \mathrm{d} x^2}$ on the edges of $\Gamma$ and either $\delta'$-type conditions or the Kirchhoff conditions at its vertices. Here $\varepsilon>0$ is a small parameter. We show that the spectrum of $\mathcal{A}_\varepsilon$ has at least $m$ gaps as $\varepsilon\to 0$ ($m\in\mathbb{N}$ is a predefined number), moreover the location of these gaps can be nicely controlled via a suitable choice of the geometry of $\Gamma$ and of coupling constants involved in $\delta'$-type conditions. - We analyze a family of singular Schr\"odinger operators describing a Neumann waveguide with a periodic array of singular traps of a $\delta'$ type. We show that in the limit when perpendicular size of the guide tends to zero and the $\delta'$ interactions are appropriately scaled, the first spectral gap is determined exclusively by geometric properties of the traps.
- Article
- Aug 1982
- RUSS MATH SURV+

CONTENTSIntroduction Chapter I. Preparatory results § 1. The cokernel of a Fredholm morphism in spaces of sections § 2. The cokernel of a Fredholm morphism in spaces of sections with bounds Chapter II. The Floquet expansion of solutions of an elliptic equation on the whole space § 1. Spaces and transformations § 2. Transformations of operators. Floquet solutions. Multipliers. Quasi-impulses § 3. Floquet expansion of solutions Chapter III. Properties of the solutions of periodic equations § 1. The disposition of multipliers and decreasing solutions § 2. Solubility of the non-homogeneous equation. Bloch's theorem § 3. Dichotomy § 4. The dispersion law Chapter IV. Other classes of periodic equations and boundary-value problems § 1. Elliptic systems § 2. Hypoelliptic equations and systems § 3. Elliptic boundary-value problems § 4. Parabolic boundary-value problems § 5. The abstract evolution equation § 6. Pseudodifferential equations § 7. Relaxation of the conditions on the smoothness of the coefficients § 8. Equations with deviating argument Comments References - Article
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- ANN HENRI POINCARE

Self-adjoint Schrödinger operators with δ and δ′-potentials supported on a smooth compact hypersurface are defined explicitly via boundary conditions. The spectral properties of these operators are investigated, regularity results on the functions in their domains are obtained, and analogues of the Birman–Schwinger principle and a variant of Krein’s formula are shown. Furthermore, Schatten–von Neumann type estimates for the differences of the powers of the resolvents of the Schrödinger operators with δ and δ′-potentials, and the Schrödinger operator without a singular interaction are proved. An immediate consequence of these estimates is the existence and completeness of the wave operators of the corresponding scattering systems, as well as the unitary equivalence of the absolutely continuous parts of the singularly perturbed and unperturbed Schrödinger operators. In the proofs of our main theorems we make use of abstract methods from extension theory of symmetric operators, some algebraic considerations and results on elliptic regularity. - ArticleFull-text available
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We describe some basic tools in the spectral theory of Schr\"odinger operator on metric graphs (also known as "quantum graph") by studying in detail some basic examples. The exposition is kept as elementary and accessible as possible. In the later sections we apply these tools to prove some results on the count of zeros of the eigenfunctions of quantum graphs. - Article
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We investigate Schr\"odinger operators with \delta- and \delta'-interactions supported on hypersurfaces, which separate the Euclidean space into finitely many bounded and unbounded Lipschitz domains. It turns out that the combinatorial properties of the partition and the spectral properties of the corresponding operators are related. As the main result we prove an operator inequality for the Schr\"odinger operators with \delta- and \delta'-interactions which is based on an optimal colouring and involves the chromatic number of the partition. This inequality implies various relations for the spectra of the Schr\"odinger operators and, in particular, it allows to transform known results for Schr\"odinger operators with \delta-interactions to Schr\"odinger operators with \delta'-interactions. - Article
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We consider a family of compact manifolds which shrinks with respect to an appropriate parameter to a graph. The main result is that the spectrum of the Laplace–Beltrami operator converges to the spectrum of the (differential) Laplacian on the graph with Kirchhoff boundary conditions at the vertices. On the other hand, if the shrinking at the vertex parts of the manifold is sufficiently slower comparing to that of the edge parts, the limiting spectrum corresponds to decoupled edges with Dirichlet boundary conditions at the endpoints. At the borderline between the two regimes we have a third possibility when the limiting spectrum can be described by a nontrivial coupling at the vertices. - Article
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We prove monotone convergence theorems for quadratic forms on a Hilbert space which improve existing results. The main tool is a canonical decomposition for any positive quadratic form h = hr + hs where hr is characterized as the largest closable form smaller than h. There is also a systematic discussion of nondensely defined forms.