Reflections on the almost Mathieu operator

An explicit derivation of a tridiagonal matrix form for the almost Mathieu operator (Harper's equation) is obtained via conjugation with a reflection operator, valid for all rational values of the rotation parameter. The difference between even and odd values of the denominator is highlighted. This tridiagonal form is useful for numerical eigenvalue computations; some Matlab code is included.This research was supported in part by an NSERC individual research grant.     

Norm estimates of almost Mathieu operators

We estimate the norm of the almost Mathieu operator , regarded as an element in the rotation C^*-algebra . In the process, we prove for every λ∈R and the inequality

     

Uniform asymptotic expansions of solutions of the Mathieu equation and the modified Mathieu equation

By the method of the model equation, uniform asymptotic expansions of the Floquet solutions of the Mathieu equation and two linearly independent solutions of the modified Mathieu equation are obtained for any real values of the separation parameter contained in these equations.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 62, pp. 60–91, 1976.     

Functional Cantor equation

We consider the class of entire functions of exponential type in relation to the scattering theory for the Schrödinger equation with a finite potential that is a finite Borel measure. These functions have a special self-similarity and satisfy q-difference functional equations. We study their asymptotic behavior and the distribution of zeros.     

Cauchy problem for the Mathieu equation away from parametric resonance

Four solutions of the Cauchy problem for Mathieu’s equation away from parametric resonance domains are analytically constructed using an asymptotic averaging method in the fourth approximation. Three solutions occur near fractional parameter values at which slow combination phases exist. The fourth solution occurs in the absence of slow phases away from parametric resonance domains and the fractional parameter values.     

Homogeneity equation almost everywhere

Summary In the paper the <InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"10"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"11"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"12"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"13"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"14"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"15"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"16"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"17"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"18"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"19"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"20"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"21"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"22"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"23"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"24"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"25"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>\phi$-homogeneity equation almost everywhere is studied. Let $G$ and $H$ be groups with zero. Assume that $(X,G)$ is a $G$-space and $(Y,H)$ is an $H$-space. We prove, under some assumption on $(Y,H)$, that if the functions $\phi\: G\to H$ and $F\: X\to Y$ satisfy the equation of $\phi$-homogeneity $F(\alpha x)\eg \phi(\alpha)F(x)$ almost everywhere in $G\times X$ then either $F$ is a zero function or there exists a homomorphism $\widetilde{\phi}\: G\to H$ such that $\phi=\widetilde{\phi}$ almost everywhere in $G$ and there exists a function $\overline{F}\: X\to Y$ such that <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[$$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation> \overline{F}(\alpha x)=\widetilde{\phi}(\alpha)\overline{F}(x) \szo{for} \alpha\in G\setminus\{0\},\quad x\in X, $$ and $F=\overline{F}$ almost everywhere in $X$.     

Asymptotic solutions and stability analysis for generalized non-homogeneous Mathieu equation

The asymptotic solutions and transition curves for the generalized form of the non-homogeneous Mathieu differential equation are investigated in this paper. This type of governing differential equation of motion arises from the dynamic behavior of a pendulum undergoing a butterfly-type end support motion. The strained parameter technique is used to obtain periodic asymptotic solutions. The transition curves for some special cases are presented and their corresponding periodic solutions with the periods of 2π and 4π are evaluated. The stability analyses of those transition curves in the ε–δ plane are carried out, analytically, using the multiple scales method. The numerical simulations for some typical points in the ε–δ plane are performed and the dynamic characteristics of the resulting phase plane trajectories are discussed.     

Absolute continuity of the integrated density of states for the almost Mathieu operator with non-critical coupling

We show that the integrated density of states of the almost Mathieu operator is absolutely continuous if and only if the coupling is non-critical. We deduce for subcritical coupling that the spectrum is purely absolutely continuous for almost every phase, settling the measure-theoretical case of Problem 6 of Barry Simon’s list of Schrödinger operator problems for the twenty-first century.     

Attenuation in the almost periodic Beverton-Holt equation

It is known that the Beverton-Holt equation with periodically varying carrying capacity has a globally attracting solution and the solution exhibits attenuation, i.e. the average of the solution over one period is strictly less than the average of the carrying capacity. Interpreted this means a periodically varying environment has a deleterious effect on the average of the solution. Also known is a randomly varying carrying capacity also yields attenuation. In this work the authors show that an almost periodic carrying capacity also yields attenuation.     

Zero-measure Cantor spectrum for Schrödinger operators with low-complexity potentials

We consider discrete one-dimensional Schrödinger operators whose potentials belong to minimal subshifts of low combinatorial complexity and prove for a large class of such operators that the spectrum is a Cantor set of zero Lebesgue measure. This is obtained through an analysis of the frequencies of the subwords occurring in the potential. Our results cover most circle map and Arnoux–Rauzy potentials.     

Calculation of solutions to the Mathieu equation and of related quantities

For the Mathieu equation, we consider finding eigenvalues with a given index (on the basis of oscillation theorems for the relevant difference equations), the stability of solutions to the difference equations, correct definition and calculation of eigenvalues and Mathieu functions with noninteger numbers, correct definition and calculation of the Mathieu characteristic exponent, and the calculation of values of solutions to the Mathieu equation for large arguments. Numerical algorithms are proposed for the problems listed above.     

Almost periodic Sturm-Liouville operators with Cantor homogeneous spectrum

To the memory of B. Ya. Levin (1906–1993) who was a teacher of our teachers and who gave us so much     

Dirichlet problem for Hermitian-Einstein equation over almost Hermitian manifold

In this paper, we investigate the Dirichlet problem for Hermitian-Einstein equation on complex vector bundle over almost Hermitian manifold, and we obtain the unique solution of the Dirichlet problem for Hermitian-Einstein equation.     

Pseudo almost periodic solutions for equation with piecewise constant argument

By using the roughness theory of exponential dichotomies and the contraction mapping, some sufficient conditions are obtained for the existence and uniqueness of pseudo almost periodic solution of the above differential equation with piecewise constant argument