Scaling for the discrete Mathieu equation

The scaling of the total width of the band for the discrete Mathieu equation is studied in the critical region near the transition between localized and extended states, for the special case in which there is one period of the modulation forp lattice spacings. A general expression for the bandwidthW in the critical region is found. At the critical point an analytic expression forpW is found which agrees to one part in 10⁸ with the result deduced from numerical work.     

Zero measure spectrum for the almost Mathieu operator

We study the almost Mathieu operator: (H _{, ,} u)(n)=u(n+1)+u(n-1)+ cos (2n+)u(n), onl ² (Z), and show that for all ,, and (Lebesgue) a.e. , the Lebesgue measure of its spectrum is precisely |4–2|. In particular, for ||=2 the spectrum is a zero measure cantor set. Moreover, for a large set of irrational 's (and ||=2) we show that the Hausdorff dimension of the spectrum is smaller than or equal to 1/2.Work partially supported by the GIF     

On the measure of the spectrum for the almost Mathieu operator

Communicated by A. Jaffe     

On the measure of the spectrum for the almost Mathieu operator

We obtain partial results on the conjecture that for the almost Mathieu operator at irrational frequency, , the measure of the spectrum,S(, , )=|4–2|. For ||2 we show that if _n is rational and irrational, then .Dedicated to Res Jost and Arthur WightmanResearch partially supported by U.S. NSF grant number DMS-8801918 and by BSF under grant number 88-00325     

On the collision operator for quantum systems with discrete spectrum

The existence of the asymptotic collision operator ψ(0) is established for a class of observables. Sufficient conditions are given for the vanishing of ψ(0) and for the existence of collisional and non-collisional invariants.     

The flavor problem and discrete symmetries

In this Letter we propose a multi-Higgs extension of the standard model with Abelian and non-Abelian discrete symmetries in which the mass matrices of the charged fermions obtained from renormalizable interactions are diagonal. However, non-diagonal contributions, that are important for obtaining the CKM matrix in the quark sector, arise from non-renormalizable dimension five interactions. Active neutrinos acquire mass only from non-renormalizable interactions, the non-diagonal entries arising through dimension five operators, while the diagonal entries comes from dimension six operators. Realistic mixing matrices in the neutrino and the quarks sectors are obtained.     

The multicritical point and the problem of coexistence of two ordered phases

Using the renormalization group theory the tetracritical point in a phase transition is associated with the coexistence of two ordered phases. The coexistence of superconductivity and magnetism and the coexistence of charge density waves and spin density waves is demonstrated.     

Anderson localization for the 1-D discrete Schrödinger operator with two-frequency potential

We prove the complete exponential localization of eigenfunctions for the 1-D discrete Schrödinger operators with quasi-periodic potentials having two basic frequencies. It is shown also that for such operators there is no forbidden zones in the spectrum, unlike the operators with one basic frequency.Dedicated to Roland Dobrushin     

Cantor Spectrum for the Almost Mathieu Operator

In this paper we use results on reducibility, localization and duality for the Almost Mathieu operator, on l ²() and its associated eigenvalue equation to deduce that for b0, ±2 and Diophantine the spectrum of the operator is a Cantor subset of the real line. This solves the so-called Ten Martini Problem for these values of b and . Moreover, we prove that for |b|0 small or large enough all spectral gaps predicted by the Gap Labelling theorem are open.     

The discrete family symmetries as the possible solution to the flavour problem

In order to explain the fermions’ masses and mixing parameters appearing in the lepton sector of the Standard Model, one proposes the extension of its symmetry. A discrete, non-Abelian subgroup of U(3) is added to the gauge group SU(3) _C × SU(2) _L × U(1) _Y . Apart from that, one assumes the existence of one extra Higgs doublet. This article focuses mainly on the mathematical theorems and computational techniques which brought us to the results.     

Counting operator analysis of the discrete spectrum of some model Hamiltonians

The first step in the counting operator analysis of the spectrum of any model Hamiltonian H is the choice of a Hermitean operator M in such a way that the third commutator with H is proportional to the first commutator. Next one calculates operators R and R^† which share some of the properties of creation and annihilation operators, and are such that M becomes a counting operator. The spectrum of H is then decomposed into multiplets, not determined by symmetries of H, but by those of a reference Hamiltonian H_ref, which is defined by H_ref=H−R−R^†, and which commutes with M. Finally, we introduce the notion of stable eigenstates. It is shown that under rather weak conditions one stable eigenstate can be used to construct another one.     

The discrete ordinates method for anisotropic scattering in the neutron transport theory: The critical sphere problem

The spherical harmonics method for anisotropic scattering in the neutron transport theory related to the critical sphere problem was investigated by Yildiz [The spherical harmonics method for anisotropic scattering in neutron transport theory: the critical sphere problem. JQSRT 2001;71:25-37]. Some numerical results and figures that they provided are incorrect. The correct numerical results for the critical radius are obtained and tabulated for different scattering parameters by using the discrete ordinates method.     

The asymptotics of the gap in the Mathieu equation

We provide a simple proof that the kth gap, Δ_k, for the Mathieu operator $\text{?d}{}^2\text{dx}{}^2\text{+ 2κ}\text{cos}\text{(2x)}$ is $\text{Δ}{}_{\mathrm{k}}\text{= 8(}\text{κ}\text{4}\text{)}{}^{\mathrm{k}}\text{[(k ? 1)!]}{}^{\mathrm{?2}}\text{(1 + o(k}{}^{\mathrm{?2}}\text{))}$, a result obtained (up to the value of an integral) by Harrell. The key observation is that what is involved is tunneling in momentum space.     

A metal-insulator transition for the almost Mathieu model

Communications in Mathematical Physics - We study the spectrum of the almost Mathieu hamiltonian: $\left( {H_x \psi } \right)\left( n \right) = \psi \left( {n + 1} \right) + \psi \left( {n - 1}...     

A model problem with the coexistence of stochastic and integrable behaviour

A one parameter family of piecewise linear measure preserving transformations of a torus which can be viewed as a perturbation of the twist mapping is introduced. Theorems on their ergodic properties for an infinite set of parameters are proved. For some parameters coexistence of stochastic and integrable behaviour is obtained.     

Exponential Dynamical Localization for the Almost Mathieu Operator

We prove that the exponential moments of the position operator stay bounded for the supercritical almost Mathieu operator with Diophantine frequency.     

Nonlinear nonautonomous discrete dynamical systems from a general discrete isomonodromy problem

Nonlinear nonautonomous discrete dynamical systems (DDS) whose continuum limits are the well-known Painlevé equations, have recently arisen in models of quantum gravity. The Painlevé equations are believed integrable because each is the isomonodromy condition for an associated linear differential equation. However, not every DDS with an integrable continuum limit is necessarily integrable. Which of the many discrete versions of the Painlevé equations inherit their integrability is not known. How to derive all their integrable discrete versions is also not known. We provide a systematic method of attacking these questions by giving a general discrete isomonodromy problem. Discrete versions of the first and second Painlevé equations are deduced from this general problem.