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Register a new userPeriodic quantum graphs with predefined spectral gaps
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Added by
Andrii Khrabustovskyi
Publication date
2020
fields
Physics
and research's language is
English
Authors
Andrii Khrabustovskyi
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Let $Gamma$ be an arbitrary $mathbb{Z}^n$-periodic metric graph, which does not coincide with a line. We consider the Hamiltonian $mathcal{H}_varepsilon$ on $Gamma$ with the action $-varepsilon^{-1}{mathrm{d}^2/mathrm{d} x^2}$ on its edges; here $varepsilon>0$ is a small parameter. Let $minmathbb{N}$. We show that under a proper choice of vertex conditions the spectrum $sigma(mathcal{H}^varepsilon)$ of $mathcal{H}^varepsilon$ has at least $m$ gaps as $varepsilon$ is small enough. We demonstrate that the asymptotic behavior of these gaps and the asymptotic behavior of the bottom of $sigma(mathcal{H}^varepsilon)$ as $varepsilonto 0$ can be completely controlled through a suitable choice of coupling constants standing in those vertex conditions. We also show how to ensure for fixed (small enough) $varepsilon$ the precise coincidence of the left endpoints of the first $m$ spectral gaps with predefined numbers.
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We consider the twisted waveguide $Omega_theta$, i.e. the domain obtained by the rotation of the bounded cross section $omega subset {mathbb R}^{2}$ of the straight tube $Omega : = omega times {mathbb R}$ at angle $theta$ which depends on the variable along the axis of $Omega$. We study the spectral properties of the Dirichlet Laplacian in $Omega_theta$, unitarily equivalent under the diffeomorphism $Omega_theta to Omega$ to the operator $H_{theta}$, self-adjoint in ${rm L}^2(Omega)$. We assume that $theta = beta - epsilon$ where $beta$ is a $2pi$-periodic function, and $epsilon$ decays at infinity. Then in the spectrum $sigma(H_beta)$ of the unperturbed operator $H_beta$ there is a semi-bounded gap $(-infty, {mathcal E}_0^+)$, and, possibly, a number of bounded open gaps $({mathcal E}_j^-, {mathcal E}_j^+)$. Since $epsilon$ decays at infinity, the essential spectra of $H_beta$ and $H_{beta - epsilon}$ coincide. We investigate the asymptotic behaviour of the discrete spectrum of $H_{beta - epsilon}$ near an arbitrary fixed spectral edge ${mathcal E}_j^pm$. We establish necessary and quite close sufficient conditions which guarantee the finiteness of $sigma_{rm disc}(H_{beta-epsilon})$ in a neighbourhood of ${mathcal E}_j^pm$. In the case where the necessary conditions are violated, we obtain the main asymptotic term of the corresponding eigenvalue counting function. The effective Hamiltonian which governs the the asymptotics of $sigma_{rm disc}(H_{beta-epsilon})$ near ${mathcal E}_j^pm$ could be represented as a finite orthogonal sum of operators of the form $-mufrac{d^2}{dx^2} - eta epsilon$, self-adjoint in ${rm L}^2({mathbb R})$; here, $mu > 0$ is a constant related to the so-called effective mass, while $eta$ is $2pi$-periodic function depending on $beta$ and $omega$.
We consider the spectrum of the almost Mathieu operator $H_alpha$ with frequency $alpha$ and in the case of the critical coupling. Let an irrational $alpha$ be such that $|alpha-p_n/q_n|<c q_n^{-varkappa}$, where $p_n/q_n$, $n=1,2,dots$ are the convergents to $alpha$, and $c$, $varkappa$ are positive absolute constants, $varkappa<56$. Assuming certain conditions on the parity of the coefficients of the continued fraction of $alpha$, we show that the central gaps of $H_{p_n/q_n}$, $n=1,2,dots$, are inherited as spectral gaps of $H_alpha$ of length at least $cq_n^{-varkappa/2}$, $c>0$.
We investigate the bottom of the spectra of infinite quantum graphs, i.e., Laplace operators on metric graphs having infinitely many edges and vertices. We introduce a new definition of the isoperimetric constant for quantum graphs and then prove the Cheeger-type estimate. Our definition of the isoperimetric constant is purely combinatorial and thus it establishes connections with the combinatorial isoperimetric constant, one of the central objects in spectral graph theory and in the theory of simple random walks on graphs. The latter enables us to prove a number of criteria for quantum graphs to be uniformly positive or to have purely discrete spectrum. We demonstrate our findings by considering trees, antitrees and Cayley graphs of finitely generated groups.
In geometric analysis, an index theorem relates the difference of the numbers of solutions of two differential equations to the topological structure of the manifold or bundle concerned, sometimes using the heat kernels of two higher-order differential operators as an intermediary. In this paper, the case of quantum graphs is addressed. A quantum graph is a graph considered as a (singular) one-dimensional variety and equipped with a second-order differential Hamiltonian H (a Laplacian) with suitable conditions at vertices. For the case of scale-invariant vertex conditions (i.e., conditions that do not mix the values of functions and of their derivatives), the constant term of the heat-kernel expansion is shown to be proportional to the trace of the internal scattering matrix of the graph. This observation is placed into the index-theory context by factoring the Laplacian into two first-order operators, H =A*A, and relating the constant term to the index of A. An independent consideration provides an index formula for any differential operator on a finite quantum graph in terms of the vertex conditions. It is found also that the algebraic multiplicity of 0 as a root of the secular determinant of H is the sum of the nullities of A and A*.
For non-critical almost Mathieu operators with Diophantine frequency, we establish exponential asymptotics on the size of spectral gaps, and show that the spectrum is homogeneous. We also prove the homogeneity of the spectrum for Schodinger operators with (measure-theoretically) typical quasi-periodic analytic potentials and fixed strong Diophantine frequency. As applications, we show the discrete version of Deifts conjecture cite{Deift, Deift17} for subcritical analytic quasi-periodic initial data and solve a series of open problems of Damanik-Goldstein et al cite{BDGL, DGL1, dgsv, Go} and Kotani cite{Kot97}.
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