We describe large classes of compact self-adjoint Hankel operators whose eigenvalues have power asymptotics and obtain explicit expressions for the coefficient in front of the leading term. The results are stated both in the discrete and continuous representations for Hankel operators. We also elucidate two key principles underpinning the proof of such asymptotic relations. We call them the localization principle and the symmetry principle . The localization principle says that disjoint components of the singular support of the symbol of a Hankel operator make independent contributions into the asymptotics of eigenvalues. The symmetry principle says that if the singular support of a symbol does not contain the points @math and @math in the discrete case (or the points @math and @math in the continuous case), then the spectrum of the corresponding Hankel operator is asymptotically symmetric with respect to the reflection around zero.
We finally mention the fundamental paper @cite_8 where the spectra of all bounded self-adjoint Hankel operators were characterized in terms of a certain balance of their positive and negative parts. In particular, the spectra of compact Hankel operators were characterized by the two conditions: (i) the multiplicities of the eigenvalues @math and @math do not differ by more than one; (ii) if the point @math is an eigenvalue, then necessarily it has infinite multiplicity. The first of these conditions is similar in spirit to the asymptotic formulas , for the eigenvalues, but of course neither of these two results implies the other one.
@cite_8
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We consider the Schrodinger operator on nanoribbons (tight-binding models) in an external electric potential V on the plane. The corresponding electric field is perpendicular to the axis of the nanoribbon. If V = 0, then the spectrum of the Schrodinger operator consists of two non-flat bands and one flat band (an eigenvalue with infinite multiplicity) between them. If we switch on a weak electric potential V → 0, then there are two cases: (1) this eigenvalue splits into the small spectral band. We determine the asymptotics of the spectral bands for small fields. (2) the unperturbed eigenvalue remains the flat band. We describe all potentials when the unperturbed eigenvalue remains the flat band and when one splits into the small band of the continuous spectrum. Comparison between formulae for the counting functions of the heights tn of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the tn are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian Hcl. Many features of Hcl are provided by the analogy; for example, the "Riemann dynamics" should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers. Statistics of the tn have a similar structure to those of the semiclassical En; in particular, they display random-matrix universality at short range, and nonuniversal behaviour over longer ranges. Very refined features of the statistics of the tn can be computed accurately from formulae with quantum analogues. The Riemann-Siegel formula for the zeta function is described in detail. Its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations. We speculate that the Riemann dynamics is related to the trajectories generated by the classical hamiltonian Hcl=XP. 1. IntroductionIn this work we address to the problem of the membership to the ideal of r-nuclearoperators for integral operators of the form(1.1) Tf(x) =Z G(x)F(xy)H(y)f(y)dy;where Gand H are bounded functions and F is a smooth function admitting asuitable Taylor expansion. A such kind of operator arises in the theory of laser,particularly in the form of the Fox-Li operator which has attracted the interest ofresearchers in recent years(cf. [BBIN10], [BGI12], [BGHI12]). In those works, theauthors were focused in the investigation of the asymptotic distribution of the sin-gular values and eigenvalues. More recently, the r-nuclearity and its connection withGrothendieck-Lidskii’s formula on compact Lie groups and applications to some dif-ferential operators have been studied in [DR13]. Applications of nuclear operators tothe study of periodic pseudo-di erential operators have been established in [DW13].The nuclearity of operators on spaces of Bochner integrable functions(cf. [DE09])and the corresponding relation with the martingale vector-maximal function for thestudy of traces have been considered in [Del]. Recent progress on the nuclearity andsingular values can also be found in [FMO09] and [MO12].The Fox-Li operator was introduced in [FL61], A. G. Fox and T. Li considered therepeated reection of an electromagnetic wave between two plane-parallel rectangularmirrors. The current density at the surface of each reector satis es the eigenvalueintegral equationTf(x) = f(x):For a good account on the physics of such kind of operators we recommend the chapterXI, section 60 in the book [NE05] by Trefethen and Embree. The Fox-Li operator is We consider the spectral problem for the random Schrodinger operator on the multidimensional lattice torus increasing to the whole of lattice, with an i.i.d. potential (Anderson Hamiltonian). We obtain the explicit almost sure asymptotic expansion formulas for the extreme eigenvalues and eigenfunctions in the intermediate rank case, provided the upper distributional tails of potential decay at infinity slower than the double exponential function. For the fractional-exponential tails (including Weibull’s and Gaussian distributions), extremal type limit theorems for eigenvalues are proved, and the strong influence of parameters of the model on a specification of normalizing constants is described. In the proof we use the finite-rank perturbation arguments based on the cluster expansion for resolvents.