Sharp spectral estimates for periodic matrix-valued Jacobi operators

For the periodic matrix-valued Jacobi operator $J$ we obtain the estimate of the Lebesgue measure of the spectrum |\s(J)|\le4 \min_n\Tr(a_na_n^*)^\frac12, where $a_n$ are off-diagonal elements of $J$. Moreover estimates of width of spectral bands are obtained.Comment: 3 page

Determinants and traces of multidimensional discrete periodic operators with defects

As it is shown in previous works, discrete periodic operators with defects are unitarily equivalent to the operators of the form $${\mathcal A}{\bf u}={\bf A}_0{\bf u}+{\bf A}_1\int_0^1dk_1{\bf B}_1{\bf u}+...+{\bf A}_N\int_0^1dk_1...\int_0^1dk_N{\bf B}_N{\bf u},\ \ {\bf u}\in L^2([0,1]^N,\mathbb{C}^M),$$ where $({\bf A},{\bf B})(k_1,...,k_N)$ are continuous matrix-valued functions of appropriate sizes. All such operators form a non-closed algebra ${\mathscr H}_{N,M}$. In this article we show that there exist a trace $\pmb{\tau}$ and a determinant $\pmb{\pi}$ defined for operators from ${\mathscr H}_{N,M}$ with the properties $$\pmb{\tau}(\alpha{\mathcal A}+\beta{\mathcal B})=\alpha\pmb{\tau}({\mathcal A})+\beta\pmb{\tau}({\mathcal B}),\ \ \pmb{\tau}({\mathcal A}{\mathcal B})=\pmb{\tau}({\mathcal B}{\mathcal A}),\ \ \pmb{\pi}({\mathcal A}{\mathcal B})=\pmb{\pi}({\mathcal A})\pmb{\pi}({\mathcal B}),\ \ \pmb{\pi}(e^{{\mathcal A}})=e^{\pmb{\tau}({\mathcal A})}.$$ The mappings $\pmb{\pi}$, $\pmb{\tau}$ are vector-valued functions. While $\pmb{\pi}$ has a complex structure, $\pmb{\tau}$ is simple $$\pmb{\tau}({\mathcal A})=\left({\rm Tr}{\bf A}_0,\int_0^1dk_1{\rm Tr}{\bf B}_1{\bf A}_1,...,\int_0^1dk_1...\int_0^1dk_N{\rm Tr}{\bf B}_N{\bf A}_N\right).$$ There exists the norm under which the closure $\overline{{\mathscr H}}_{N,M}$ is a Banach algebra, and $\pmb{\pi}$, $\pmb{\tau}$ are continuous (analytic) mappings. This algebra contains simultaneously all operators of multiplication by matrix-valued functions and all operators from the trace class. Thus, it generalizes the other algebras for which determinants and traces was previously defined

Mixed multidimensional integral operators with piecewise constant kernels and their representations

We consider the algebra of mixed multidimensional integral operators. In particular, Fredholm integral operators of the first and second kind belongs to this algebra. For the piecewise constant kernels we provide an explicit representation of the algebra as a product of simple matrix algebras. This representation allows us to compute the inverse operators (or to solve the corresponding integral equations) and to find the spectrum explicitly. Moreover, explicit traces and determinants are also constructed. So, roughly speaking, the analysis of integral operators is reduced to the analysis of matrices. All the qualitative characteristics of the spectrum are preserved since only the kernels are approximated

Application of matrix-valued integral continued fractions to spectral problems on periodic graphs

We show that spectral problems for periodic operators on lattices with embedded defects of lower dimensions can be solved with the help of matrix-valued integral continued fractions. While these continued fractions are usual in the approximation theory, they are less known in the context of spectral problems. We show that the spectral points can be expressed as zeroes of determinants of the continued fractions. They are also useful in the study of inverse problems (one-to-one correspondence between spectral data and defects). Finally, the explicit formula for the resolvent in terms of the continued fractions is also provided. We apply some of our results to the Schr\"odinger operator acting on the graphene with line and point defects

Recovery of defects from the information at detectors

The discrete wave equation in a multidimensional uniform space with local defects and sources is considered. The characterization of all possible defect configurations corresponding to given amplitudes of waves at the receivers (detectors) is provided.Comment: in Inverse Problems, 055005, 201

On the measure of the spectrum of direct integrals

We obtain the estimate of the Lebesgue measure of the spectrum for the direct integral of matrix-valued functions. These estimates are applicable for a wide class of discrete periodic operators. For example: these results give new and sharp spectral bounds for 1D periodic Jacobi matrices and 2D discrete periodic Schrodinger operators

A note on sharp spectral estimates for periodic Jacobi matrices

The spectrum of three-diagonal self-adjoint $p$-periodic Jacobi matrix with positive off-diagonal elements $a_n$ an real diagonal elements $b_n$ consist of intervals separated by $p-1$ gaps $\gamma_i$, where some of the gaps can be degenerated. The following estimate is true $$\sum_{i=1}^{p-1}|\gamma_i|\geq\max(\max(4(a_1...a_p)^{\frac1p},2\max a_n)-4\min a_n,\max b_n-\min b_n).$$ We show that for any $p\in\mathbb{N}$ there are Jacobi matrices of minimal period $p$ for which the spectral estimate is sharp. The estimate is sharp for both: strongly and weakly oscillated $a_n$, $b_n$. Moreover, it improves some recent spectral estimates

Finite PDEs and finite ODEs are isomorphic

The standard view is that PDEs are much more complex than ODEs, but, as will be shown below, for finite derivatives this is not true. We consider the $C^*$-algebras ${\mathscr H}_{N,M}$ consisting of $N$-dimensional finite differential operators with $M\times M$-matrix-valued bounded periodic coefficients. We show that any ${\mathscr H}_{N,M}$ is $*$-isomorphic to the universal uniformly hyperfinite algebra (UHF algebra) $\bigotimes_{n=1}^{\infty}\mathbb{C}^{n\times n}.$ This is a complete characterization of the differential algebras. In particular, for different $N,M\in\mathbb{N}$ the algebras ${\mathscr H}_{N,M}$ are topologically and algebraically isomorphic to each other. In this sense, there is no difference between multidimensional matrix valued PDEs ${\mathscr H}_{N,M}$ and one-dimensional scalar ODEs ${\mathscr H}_{1,1}$. Roughly speaking, the multidimensional world can be emulated by the one-dimensional one

Divergence of the logarithm of a unimodular monodromy matrix near the edges of the Brillouin zone

A first-order differential system with matrix of periodic coefficients $Q(y)=Q(y+T)$ is studied for time-harmonic elastic waves in a unidirectionally periodic medium, for which the monodromy matrix $M(\omega)$ implies a propagator of the wave field over a period. The main interest in the matrix logarithm $\ln M(\omega)$ is due to the fact that it yields the 'effective' matrix $Q_{eff}(\omega)$ of the dynamic-homogenization method. For the typical case of a unimodular matrix $M(\omega)$ ($\det M=1$), it is established that the components of $\ln M(\omega)$ diverge as $(\omega -\omega_0)^{-1/2}$ with $\omega \to \omega_0,$ where $\omega_0$ is the set of frequencies of the passband/stopband crossovers at the edges of the first Brillouin zone. The divergence disappears for a homogeneous medium. Mathematical and physical aspects of this observation are discussed. Explicit analytical examples of $Q_{eff}(\omega)$ and of its diverging asymptotics at $\omega \to \omega_0$ are provided for a model of scalar waves in a two-component periodic structure. The case of high contrast due to stiff/soft layers or soft springs is elaborated. Special attention in this case is given to the asymptotics of $Q_{eff}(\omega)$ near the first stopband that occurs at the Brillouin-zone edge at arbitrary low frequency. The link to the quasi-static asymptotics of the same $Q_{eff}(\omega)$ near the point $\omega=0$ is also elucidated.Comment: 20 pages, 1 figur

Using SDO/HMI magnetograms as a source of the solar mean magnetic field data

The solar mean magnetic field (SMMF) provided by the Wilcox Solar Observatory (WSO) is compared with the SMMF acquired by the \textit{Helioseismic and Magnetic Imager} (HMI) onboard the \textit{Solar Dynamic Observatory} (SDO). We found that despite the different spectral lines and measurement techniques used in both instruments the Pearson correlation coefficient between these two datasets equals 0.86 while the conversion factor is very close to unity: B(HMI) = 0.99(2)B(WSO). We also discuss artifacts of the SDO/HMI magnetic field measurements, namely the 12 and 24-hour oscillations in SMMF and in sunspots magnetic fields that might be caused by orbital motions of the spacecraft. The artificial harmonics of SMMF reveal significant changes in amplitude and the nearly stable phase. The connection between the 24-hour harmonic amplitude of SMMF and the presence of sunspots is examined. We also found that opposite phase artificial 12 and/or 24-hour oscillations exist in sunspots of opposite polarities.Comment: 13 pages, 9 figure