We study a matrix-valued Schrödinger operator with random point interactions. We prove the absence of absolutely continuous spectrum for this operator by proving that away from a discrete set its Lyapunov exponents do not vanish. For this we use a criterion by Gol'dsheid and Margulis and we prove the Zariski denseness, in the symplectic group, of the group generated by the transfer matrices. Then we prove estimates on the transfer matrices which lead to the Hölder continuity of the Lyapunov exponents. After proving the existence of the integrated density of states of the operator, we also prove its Hölder continuity by proving a Thouless formula which links the integrated density of states to the sum of the positive Lyapunov exponents.
We establish semiclassical resolvent estimates for Schrödinger operators with long-range matrix-valued potentials. As an application we prove resonance free domains both in trapping and non-trapping situations. Our results generalize the well-known results of \cite{Burq1, Martinez1} in the case of scalar Schrödinger operators.
We prove a general Borg-type inverse spectral result for a reflectionless unitary CMV operator (CMV for Cantero, Moral, and Velázquez) associated with matrix-valued Verblunsky coefficients. More precisely, we find an explicit formula for the Verblunsky coefficients of a reflectionless CMV matrix whose spectrum consists of a connected arc on the unit circle. This extends a recent result on CMV operators with scalar-valued coefficients. In the course of deriving the Borg-type result we also use exponential Herglotz representations of Caratheodory matrix-valued functions to prove an infinite sequ…
Factorization for the matrix-valued general Jacobi system on the full-line lattice
University of Texas at Arlington（德克萨斯大学阿灵顿分校）；Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo（米却肯圣尼古拉斯德伊达尔戈大学物理与数学研究所）；National Technical University of Athens（雅典国立技术大学）；Recep Tayyip Erdogan University（雷杰普·塔伊普·埃尔多安大学）
The principal purpose of this note is to provide a reconstruction procedure for distributional matrix-valued potential coefficients of Schrödinger-type operators on a half-line from the underlying Weyl-Titchmarsh function.
We prove local and global versions of Borg-Marchenko-type uniqueness theorems for half-lattice and full-lattice CMV operators (CMV for Cantero, Moral, and Velazquez) with matrix-valued Verblunsky coefficients. While our half-lattice results are formulated in terms of matrix-valued Weyl-Titchmarsh functions, our full-lattice results involve the diagonal and main off-diagonal Green's matrices. We also develop the basics of Weyl-Titchmarsh theory for CMV operators with matrix-valued Verblunsky coefficients as this is of independent interest and an essential ingredient in proving the corresponding…
We define and examine certain matrix-valued multiplicative functionals with local Kato potential terms and use probabilistic techniques to prove that the semigroups of the corresponding partial differential operators with matrix-valued coefficients are spatially continuous and have a jointly continuous integral kernel. These partial differential operators include Yang-Mills type Hamiltonians and Pauli type Hamiltonians, with "electrical" potentials that are elements of the matrix-valued local Kato class.
We apply a relation between matrix-valued complete Bernstein functions and matrix-valued Stieltjes functions to prove that certain convolution equations for matrix-valued functions have unique solutions in a special class of functions. In particular the cases of the viscoelastic duality theorem and the Sonine equation are discussed, with applications in anisotropic linear viscoelasticity and a generalization of fractional calculus.
We study a continuous matrix-valued Anderson-type model. Both leading Lyapunov exponents of this model are proved to be positive and distinct for all ernergies in $(2,+\infty)$ except those in a discrete set, which leads to absence of absolutely continuous spectrum in $(2,+\infty)$. This result is an improvement of a previous result with Stolz. The methods, based upon a result by Breuillard and Gelander on dense subgroups in semisimple Lie groups, and a criterion by Goldsheid and Margulis, allow for singular Bernoulli distributions.
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.
A formulation of the exponential matrix solution of the matrix-valued Bratu equation is given, based on the structure of the symmetric domain of type BDI. Moreover, an analog for the symmetric domain of type CI is given.
Borg-type uniqueness theorems for matrix-valued Jacobi operators H and supersymmetric Dirac difference operators D are proved. More precisely, assuming reflectionless matrix coefficients A, B in the self-adjoint Jacobi operator H=AS^+ + A^-S^- + B (with S^\pm the right/left shift operators on the lattice Z) and the spectrum of H to be a compact interval [E_-,E_+], E_- < E_+, we prove that A and B are certain multiples of the identity matrix. An analogous result which, however, displays a certain novel nonuniqueness feature, is proved for supersymmetric self-adjoint Dirac difference operators D…
A general family of matrix valued Hermite type orthogonal polynomials is introduced and studied in detail by deriving Pearson equations for the weight and matrix valued differential equations for these matrix polynomials. This is used to derive Rodrigues formulas, explicit formulas for the squared norm and to give an explicit expression of the matrix entries as well to derive a connection formula for the matrix polynomials of Hermite type. We derive matrix valued analogues of Burchnall formulas in operational form as well explicit expansions for the matrix valued Hermite type orthogonal polyno…
We give a CLR type bound on the number of bound states of Schroedinger operators with matrix-valued potentials using the functional integral method of Lieb. This significantly improves the constant in this inequality obtained earlier by Hundertmark.
A series of physically motivated operations appearing in the study of composite materials are interpreted in terms of elementary continued fraction transforms of matrix valued, rational Stieltjes functions.
The matrix valued analog of the Euler's hypergeometric differential equation was introduced by Tirao in \cite{T2}. This equation arises in the study of matrix valued spherical functions and in the theory of matrix valued orthogonal polynomials. The goal of this paper is to extend naturally the number of parameters of Tirao's equation in order to get a generalized matrix valued hypergeometric equation. We take advantage of the tools and strategies developed in \cite{T2} to identify the corresponding matrix hypergeometric functions ${}_nF_m$. We prove that, if n=m+1, this functions are analytic …
Matrix-valued Cauchy bi-orthogonal polynomials were proposed in this paper, together with its quasideterminant expression. It is shown that the coefficients in four-term recurrence relation for matrix-valued Cauchy bi-orthogonal polynomials should satisfy a novel noncommutative integrable system, whose Lax pair is given by fractional differential operators with non-abelian variables.
We consider discrete one-dimensional random Schroedinger operators with decaying matrix-valued, independent potentials. We show that if the l^2-norm of this potential has finite expectation value with respect to the product measure then almost surely the Schroedinger operator has an interval of purely absolutely continuous (ac) spectrum. We apply this result to Schroedinger operators on a strip. This work provides a new proof and generalizes a result obtained by Delyon, Simon, and Souillard.
The article is devoted to the study of $R$-matrix-valued Lax pairs for $N$-body (elliptic) Calogero-Moser models. Their matrix elements are given by quantum ${\rm GL}_{\tilde N}$ $R$-matrices of Baxter-Belavin type. For $\tilde N=1$ the widely known Krichever's Lax pair with spectral parameter is reproduced. First, we construct the $R$-matrix-valued Lax pairs for Calogero-Moser models associated with classical root systems. For this purpose we study generalizations of the D'Hoker-Phong Lax pairs. It appeared that in the $R$-matrix-valued case the Lax pairs exist in special cases only. The numb…
The main purpose of this paper is to obtain an explicit expression of a family of matrix valued orthogonal polynomials {P_n}_n, with respect to a weight W, that are eigenfunctions of a second order differential operator D. The weight W and the differential operator D were found in [12], using some aspects of the theory of the spherical functions associated to the complex projective spaces. We also find other second order differential operator E symmetric with respect to W and we describe the algebra generated by D and E.
We provide a comprehensive analysis of matrix-valued Herglotz functions and illustrate their applications in the spectral theory of self-adjoint Hamiltonian systems including matrix-valued Schrödinger and Dirac-type operators. Special emphasis is devoted to appropriate matrix-valued extensions of the well-known Aronszajn-Donoghue theory concerning support properties of measures in their Nevanlinna-Riesz-Herglotz representation. In particular, we study a class of linear fractional transformations M_A(z) of a given n \times n Herglotz matrix M(z) and prove that the minimal support of the absolut…
Consider the Laguerre polynomials and deform them by the introduction in the measure of an exponential singularity at zero. In [Chen Y., Its A., J. Approx. Theory 162 (2010), 270-297, arXiv:0808.3590] the authors proved that this deformation can be described by systems of differential/difference equations for the corresponding recursion coefficients and that these equations, ultimately, are equivalent to the Painlevé III equation and its Bäcklund/Schlesinger transformations. Here we prove that an analogue result holds for some kind of semiclassical matrix-valued orthogonal polynomials of Lague…
We derive new characterisations of the matrix $\mathrmΦ$-entropy functionals introduced in [Electron.~J.~Probab., 19(20): 1--30, 2014]. Notably, all known equivalent characterisations of the classical $Φ$-entropies have their matrix correspondences. Next, we propose an operator-valued generalisation of the matrix $Φ$-entropy functionals, and prove their subadditivity under Löwner partial ordering. Our results demonstrate that the subadditivity of operator-valued $Φ$-entropies is equivalent to the convexity of various related functions. This result can be used to demonstrate an interesting resu…
We study, both analytically and numerically, the Boltzmann transport equation for the Hubbard chain with nearest neighbor hopping and spatially homogeneous initial condition. The time-dependent Wigner function is matrix-valued because of spin. The H-theorem holds. The nearest neighbor chain is integrable which, on the kinetic level, is reflected by infinitely many additional conservation laws and linked to the fact that there are also non-thermal stationary states. We characterize all stationary solutions. Numerically, we observe an exponentially fast convergence to stationarity and investigat…
In this paper we discuss $R$-matrix-valued Lax pairs for ${\rm sl}_N$ Calogero-Moser model and their relation to integrable quantum long-range spin chains of the Haldane-Shastry-Inozemtsev type. First, we construct the $R$-matrix-valued Lax pairs for the third flow of the classical Calogero-Moser model. Then we notice that the scalar parts (in the auxiliary space) of the $M$-matrices corresponding to the second and third flows have form of special spin exchange operators. The freezing trick restricts them to quantum Hamiltonians of long-range spin chains. We show that for a special choice of t…
