Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Spectral Asymptotics for Waveguides with Perturbed Periodic Twisting   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 $2\pi$-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 $-\mu\frac{d^2}{dx^2} - \eta
\epsilon$, self-adjoint in ${\rm L}^2({\mathbb R})$; here, $\mu > 0$ is a
constant related to the so-called effective mass, while $\eta$ is
$2\pi$-periodic function depending on $\beta$ and $\omega$.

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The categories of this paper are: dg-ga
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The categories of this paper are: hep-ex
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The categories of this paper are: atom-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: solv-int
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The categories of this paper are: q-bio
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The categories of this paper are: plasm-ph
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The categories of this paper are: supr-con
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The categories of this paper are: hep-lat
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The categories of this paper are: nucl-th
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The categories of this paper are: q-fin
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The categories of this paper are: cs
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The categories of this paper are: cond-mat
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The categories of this paper are: nlin
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The categories of this paper are: mtrl-th
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The categories of this paper are: physics