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Register a new userSpectral asymptotics and Lame spectrum for coupled particles in periodic potentials
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We make two observations on the motion of coupled particles in a periodic potential. Coupled pendula, or the space-discretized sine-Gordon equation is an example of this problem. Linearized spectrum of the synchronous motion turns out to have a hidden asymptotic periodicity in its dependence on the energy; this is the gist of the first observation. Our second observation is the discovery of a special property of the purely sinusoidal potentials: the linearization around the synchronous solution is equivalent to the classical Lam`e equation. As a consequence, {it all but one instability zones of the linearized equation collapse to a point for the one-harmonic potentials}. This provides a new example where Lames finite zone potential arises in the simplest possible setting.
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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}.
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 $omega$.
In this article, we present a new approach to averaging in non-Hamiltonian systems with periodic forcing. The results here do not depend on the existence of a small parameter. In fact, we show that our averaging method fits into an appropriate nonlinear equivalence problem, and that this problem can be solved formally by using the Lie transform framework to linearize it. According to this approach, we derive formal coordinate transformations associated with both first-order and higher-order averaging, which result in more manageable formulae than the classical ones. Using these transformations, it is possible to correct the solution of an averaged system by recovering the oscillatory components of the original non-averaged system. In this framework, the inverse transformations are also defined explicitly by formal series; they allow the estimation of appropriate initial data for each higher-order averaged system, respecting the equivalence relation. Finally, we show how these methods can be used for identifying and computing periodic solutions for a very large class of nonlinear systems with time-periodic forcing. We test the validity of our approach by analyzing both the first-order and the second-order averaged system for a problem in atmospheric chemistry.
The spectrum of the non-self-adjoint Zakharov-Shabat operator with periodic potentials is studied, and its explicit dependence on the presence of a semiclassical parameter in the problem is also considered. Several new results are obtained. In particular: (i) it is proved that the resolvent set has two connected components, (ii) new bounds on the location of the Floquet and Dirichlet spectra are obtained, some of which depend explicitly on the value of the semiclassical parameter, (iii) it is proved that the spectrum localizes to a cross in the spectral plane in the semiclassical limit. The results are illustrated by discussing several examples in which the spectrum is computed analytically or numerically.
We study the asymptotics in n for n-dimensional Toeplitz determinants whose symbols possess Fisher-Hartwig singularities on a smooth background. We prove the general non-degenerate asymptotic behavior as conjectured by Basor and Tracy. We also obtain asymptotics of Hankel determinants on a finite interval as well as determinants of Toeplitz+Hankel type. Our analysis is based on a study of the related system of orthogonal polynomials on the unit circle using the Riemann-Hilbert approach.
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