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

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

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

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

Borg type uniqueness Theorems for periodic Jacobi operators with matrix valued coefficients

We give a simple proof of Borg type uniqueness Theorems for periodic Jacobi operators with matrix valued coefficients

Lyapunov functions for periodic matrix-valued Jacobi operators

We consider periodic matrix-valued Jacobi operators. The spectrum of this operator is absolutely continuous and consists of intervals separated by gaps. We define the Lyapunov function, which is analytic on an associated Riemann surface. On each sheet the Lyapunov function has the standard properties of the Lyapunov function for the scalar case. We show that this function has (real or complex) branch points, which we call resonances. We prove that there exist two types of gaps: i) stable gaps, i.e., the endpoints are periodic and anti-periodic eigenvalues, ii) unstable (resonance) gaps, i.e., the endpoints are resonances (real branch points). We show that some spectral data determine the spectrum (counting multiplicity) of the Jacobi operator