IWOTA 2019
International Workshop on
Operator Theory and its Applications
IWOTA 2019
International Workshop on
Operator Theory and its Applications
In this talk, we are interested in the inverse problem for the biharmonic equation posed on a rectangle, which is of great importance in many areas of industry and engineering. We show that the problem under consideration is ill-posed, therefore, to solve it, we opted for a regularization method via Multiparameter regularization method. The numerical implementation is based on the application of the semi-discrete finite difference method for a sequence of well-posed direct problems depending on a two small parameters of regularization.
Numerical results are performed for a rectangle domain showing the effectiveness of the proposed method
We consider the Dirichlet Laplacian in a two-dimensional strip composed of segments translated along a straight line with respect to a rotation angle with velocity diverging at infinity. We show that this model exhibits a "raise of dimension" at infinity leading to an essential spectrum determined by an asymptotic three-dimensional tube of annular cross section. If the cross section of the asymptotic tube is a disc, we also prove the existence of discrete eigenvalues below the essential spectrum.
Joint work with David Krejcirik (Prague).
In the paper various cases of dependency on eigenparameter of one of the boundary conditions of a Sturm-Liouville problem were discussed. The attention was concentrated on the cases of affine, quadratic, and rational dependencies. The basis properties of the eigenspace consisted of eigenvectors and some associated eigenvectors were discussed. It is shown that when there are some associated eigenvectors in the space then its basis properties heavily depend on the choice of associated vectors. The study was also supplied with interesting examples. The use of mathematical software Maple for abstract calculations was also demonstrated.
We consider the non-self-adjoint third order differential operator on the circle, where the coefficients are real. The Lax equation for this operator is equivalent to the good Boussinesq equation on the circle. We construct the $3$-sheeted Riemann surface for the multipliers. Ramifications of this surface are invariant with respect to the Boussinesq flow. Moreover, we consider the three-point problem for this operator on the interval $[0,2]$ with the Dirichlet conditions at the points $0$, $1$ and $2$. The eigenvalues of this problem consist an auxiliary spectrum for the inverse spectral problem associated with the good Boussinesq equation. We prove that the large ramifications and the large three-point eigenvalues are real. We determine the high energy asymptotics of the ramifications and the three-point eigenvalues. We also determine the trace formulas in terms of the ramifications and the three-point eigenvalues.
This is a joint work with Prof E. Korotyaev from S. Petersburg State University.
A bounded linear operator $T:H_1\rightarrow H_2$, where $H_1, H_2$ are Hilbert spaces, is called norm attaining if there exist $x\in H_1$ with unit norm such that $\|Tx\|=\|T\|$. If for every closed subspace $M\subseteq H_1$, the operator $T|_M:M\rightarrow H_2$ is norm attaining, then $T$ is called absolutely norm attaining. If in the above definitions $\|T\|$ is replaced by the minimum modulus, $m(T):=\inf\{\|Tx\|:x\in H_1,\|x\|=1\}$, then $T$ is called minimum attaining and absolutely minimum attaining, respectively.
In this article, we show the existence of a non-trivial hyperinvariant subspace for absolutely norm (minimum) attaining normaloid (minimaloid) operators.
We investigate spectral properties of the operator describing a quantum particle confined to a planar domain $\Omega$ rotating around a fixed point with an angular velocity $\omega$ and demonstrate several properties of its principal eigenvalue $\lambda_1^\omega$.
The logarithmic residue theorem from complex function theory gives a connection between the number of zeros of an analytic function $f$ in a domain $D$ and the contour integral of the logarithmic derivative $f^\prime / f$ over the boundary of $D$. What is the situation when one considers analytic functions having their values in target Banach algebras more general than the complex plane? The issue involves the study of sums of idempotents and has surprisingly many ramifications. Two special cases will be singled out for a more detailed discussion. The first is the commutative situation where sophisticated elements from Gelfand Theory play a role. The second is concerned with a case which is archetypical for a variety of matrix algebras, namely the ones determined by a zero pattern given by a reflexive, transitive digraph. The algebra in question is that of the block upper triangular matrices. Handling it requires elements from Integer Programming and the celebrated Farkas Lemma.
The talk reports on joint work with Torsten Ehrhardt (Santa Cruz, California) and Bernd Silbermann (Chemnitz, Germany). It is also based on earlier work with Albert Wagelmans (Rotterdam, The Netherlands).
Is there a general method to approximate spectral properties of a given operator? If so, can we control the approximations and which spectral properties are preserved? These questions are addressed during this talk. We will put a special focus on operators defined via an underlying dynamics and geometry. In this case, we study how deformations of our dynamical system respectively its geometry lead to suitable approximations.
We consider symmetric operator $A$ on a Hilbert space $H$ which is uniraily equivalent to its scalar multiple, that is $UA=qAU$, where $q \gt 1$. We investigate existence of a self-adjoint extensions $\hat A$ of the operator $A$ which are also scale-invariant. Examples of scale-invariant differential and difference operators are given.
A generalization of the notion of $A$-frame [3] to the continuous setting and for a densely defined and possibly unbounded operator $A$ on a Hilbert space $\mathcal{H}$ (with domain $\mathcal{D}(A)$) has been introduced and studied in [1]. Equivalent formulations in terms of atomic systems, existence results and some characterizations are given.
If the operator $A$ is bounded in $\mathcal{H}$, then every element $Af$ in $\mathcal{R}(A)$, the range of $A$, can be decomposed as a combination of a family of vectors (the elements of an $A$-frame, which are a Bessel family and do not necessarily belong to $\mathcal{R}(A)$) with coefficients continuously depending on $f$ (see e.g. [3] in the discrete case and [4] in the continuous one). On the contrary, the unboundedness of $A$ leads to the fact that, in a similar decomposition of the elements in $\mathcal{R}(A)$, the coefficients can not depend continuously on $f$. In [1] this problem is addressed in two ways, going over what have been done in the discrete case in [2]. In one case, a non-Bessel family and coefficients depending continuously on $ f\in\mathcal{D}(A)$ have been considered, in another one we take a Bessel family and coefficients depending continuously on $f\in\mathcal{D}(A)$ only in the graph topology of $A$.
Furthermore, a part of the study has been devoted to the analysis, synthesis and frame operators of some continuous families of vectors in $\mathcal{H}$, which are not bounded operators anymore, neither necessarily densely defined.
We study linear perturbations of Donoghue classes of scalar Herglotz-Nevanlinna functions by a real parameter $Q$ and their representations as impedance of conservative L-systems. Perturbation classes $\mathfrak{M}^Q$, $\mathfrak{M}^Q_\kappa$, $\mathfrak{M}^{-1,Q}_\kappa$ are introduced and for each class the realization theorem is stated and proved. We use a new approach that leads to explicit new formulas describing the von Neumann parameter of the main operator of a realizing L-system and the unimodular one corresponding to a self-adjoint extension of the symmetric part of the main operator. The dynamics of the presented formulas as functions of $Q$ is obtained. Examples with differential operators that illustrate the obtained results are presented.
The talk is based on joint work with E. Tsekanovskiĭ.
In this talk, we introduce the B-discrete spectrum of an unbounded closed operator. Then we prove that a closed operator has a purely B-discrete spectrum if and only if it has a meromorphic resolvent. An illustrating example of operator with purely B-discrete spectrum, is given by the Schrödinger operator with a constant magnetic field.
A quantum graph consisting of a ring and coupled infinite wires is considered. The Rashba Hamiltonian with spin-orbit interaction on edges and the Kirchhoff coupling condition at the vertices are assumed. The scattering matrix is obtained. Completeness of the system of resonance states on the ring is studied. The proof technique is based on the factorization theorems for the characteristic function in the Sz-Nagy functional model related to the scattering matrix for the quantum graph. Namely, the criterion of the absence of a singular inner factor is used. The problem of persistent current for coupled rings is studied.
The work was partially supported by Russian Science Foundation, grant 16-11-10330.
Joint work with Igor Popov, Maria Smolkina (ITMO University).
First order asymptotics of fourth order differential operators with eigenvalue dependent boundary conditions and periodic boundary conditions. The quadratic operator polynomials have self-adjoint coefficients.
A general common fixed point theorem for two pairs of weakly subsequentially continuous mappings (recently introduced) satisfying a significant estimated implicit function is proved. An extension of this result is thereby obtained. Our results assert the existence and uniqueness of common fixed points in several cases.
We examine the spectrum of the operator $-A d^2/dx^2+ B x^2$ acting on a suitable domain of $L^2(\mathbb R)^n$ where $A$ and $B$ are $n\times n$ matrices. This is a non-comutative version of the quantum mechanical harmonic oscillator, which turns out to exhibit highly non-trivial spectral properties. We will announce new results as well as open problems. Joint work in collaboration with Anastasia Doiku.
We shall show that for a time harmonic Maxwell equation the coefficients may be determined by the impedance map determined on a smooth subset of the boundary
In this talk, we discuss sectoriality of the form-sum of the form associated with a sectorial diagonal operator and a form associated with not necessarily square-summable functions $f$ and $g$.
This is based on the joint work with Rajeev Gupta and K. B. Sinha.
The Riesz-Dunford functional calculus, and its extensions to unbounded operators, allow us to define functions of operators such as the fractional powers of a linear operator like the Laplacian. One of the reasons to replace the Laplace operator in the heat equation, by its fractional powers is that the new fractional equation has solutions with finite speed propagation of the heat. This is equivalent, in some cases, to a modification of the Fourier law in the case of homogeneous materials. In the case of non homogeneous material we have to consider more general Fourier's law defined by vector operators with suitable nonconstant coefficients that multiply the spatial derivatives. It is possible to compute directly the fractional powers of vector operators using a new functional calculus, the so called $S$-functional calculus, based on the notion of $S$-spectrum, that has been developed in more recent times. In fact, the main problem to develop a spectral theory for vector operators was to understand the natural notion of spectrum for quaternionic linear operators that contain as a particular case vector operators like the gradient operator or its generalizations. This problem was solved only in 2006 with the discovery of the $S$-spectrum for quaternionic linear operators and since then the quaternionic spectral theory has rapidly grown. In this talk we present the applications of the spectral theory on the S-spectrum to fractional diffusion problems. For a systematic treatment of this theory and its applications see the books [1,2,3,4].
Courant’s nodal domain theorem provides a natural generalization of Sturm-Liouville theory to higher dimensions; however, the result is in general not sharp. It was recently shown that the nodal deficiency of an eigenfunction is encoded in the spectrum of the Dirichlet-to-Neumann operators for the eigenfunction's positive and negative nodal domains. While originally derived using symplectic methods, this result can also be understood through the spectral flow for a family of boundary conditions imposed on the nodal set. In this talk I will describe this flow for a Schrödinger operator with separable potential on a rectangular domain, and describe a mechanism by which low energy eigenfunctions do or do not contribute to the nodal deficiency. Operators on non-rectangular domains and quantum graphs will also be discussed. This talk represents joint work with Gregory Berkolaiko (Texas A&M) and Jeremy Marzuola (UNC Chapel Hill).
Developments in the field of quantum mechanics originate from the basics of Quantum theory. Introduction of the concept of quantum and subsequent initiation of quantum theory by Max Planck in the year 1900 is regarded as the dividing point between the classical physics and the modern (quantum) physics. In Quantum mechanics, states denote the elements, whereas the self adjoint operators are called observables. The concept is based on the consideration of space L2(-???). These basics led to the research in the field of operators, especially the self adjoint operators and non self adjoint operators over Hilbert space.
In year 1909, H. Weyl [50], concluded that an ‘essential’ part of the spectrum is left invariant when a self adjoint operator is perturbed by a self adjoint compact operator in a Hilbert space. The invariant part specifically contains the limit points of spectrum as well as the points of infinite multiplicity. As an acknowledgement of the contribution by the physicist, this spectrum was later on termed as Weyl spectrum. This observation became the classic version of Weyl’s theorem. As per this version, the compliment of bounded self adjoint operators coincides with the isolated points of the spectrum, which are the eigen values of finite multiplicity. In the present version, a bounded linear operator is said to satisfy Weyl’s theorem if the compliment of the Weyl’s spectrum in the spectrum equals the set of isolated eigenvalues of finite multiplicity.
Ever since its formulation, for its large number of applications to various branches of applied mathematics, the concept of Weyl spectrum has been a subject of research for a host of mathematicians all over the globe.
Drawing motivation from the definitions given by Schechter in 1965 and L. A. Coburn in 1966, the purpose of the talk is to present different manners to introduce α-Weyl operators and α-Weyl Spectrum using the very first investigations done by G. Edgar, J. Ernest and S. G. Lee to introduce $\alpha$-Fredholm operators by defining $\alpha$-closed subspace.
Researches related to the free-space quantum information transfer are very important now. The main reason is that the property of quantum information units to be entangled provides absolute security of data transmission. We study the ability of an atmospheric quantum channel to preserve the entanglement during the transmission of information encoded by the modes of the Gaussian beam. The operator approach is used. Namely, to estimate the quality of the entanglement transmission we find the distance between the transformation matrix of a quantum channel and the subspace of matrices being tensor products of matrices corresponding to separate qubits. The technique is related to singular expansions for matrices. It is shown that if the mode numbers increase, the quantum channel retains the entanglement better, that is, the probability of transmission error decreases.
In this talk I will report on spectral triples associated to finite, higher-rank graph $C^\ast$-algebras and the relation between eigenfunctions of the associated Laplacian and $k$-graph wavelets.
This is joint work with E. Gillaspy, A. Julien, S. Kang, and J. Packer.
We present a scheme for using the method of boundary triples to approach the spectral theory of powers of the classical hypergeometric differential equation. We focus our attention on the special case of the $n$-th power of the Jacobi differential equation and present the associated Weyl $m$-function, a $2n\times 2n$ matrix-valued Nevanlinna-Herglotz function, for self-adjoint extensions. Comments will be made on a generalization to powers of general Sturm-Liouville differential equations. The talk is based on joint work with Annemarie Luger (Stockholm University).
Let $H$ be a positive self-adjoint elliptic operator and $t>0$. The associated operators $K=e^{-tH}$, $T=e^{-t\sqrt{H}}$, $U=e^{-itH}$,... and their integral kernels differ widely in analytical properties and physical significance but are all related. Their asymptotics at small $t$ reflect the suitably averaged asymptotics at large $\omega$ of the eigenvalue distribution $N(\omega^2)$ of $H$. $T$ provides information about vacuum energy in quantum field theory. The asymptotic series of its kernel contains nonlocal information about $H$ that is not visible in the series of $K$. However, local terms that do correlate with those heat-kernel terms produce formally infinite terms in the energy that must be sstematically argued away ("renormalization"). We are studying a model of a scalar field interacting with a "soft wall" described by a power-law function, $V= z^\alpha$ for $z\gt 0$. The contributions of most normal modes can be accurately calculated by WKB (phase-integral) asymptotics, and thereby the divergent energy structure predicted from the heat kernel series is reproduced. The physical meaning of the infinite terms is revealed by treating the potential $V$ as a dynamical field itself and observing that all the divergent terms can be identified as renormalizations of "bare" coupling constants connecting the two fields. The remaining, finite part of the energy, the part of greatest ultimate physical interest, requires numerical computation in the regime of small $\omega$ and $z$.
In this talk, we will consider the non-homogeneous, ill-posed Cauchy problem $du/dt=Au+h(t)$, $0\lt t \lt T$, $u(0)=x$, where $-A$ generates a holomorphic semigroup on $X$. While several methods of regularization have been applied to the problem, often calculations rely acutely on the value of the angle of the semigroup. We will investigate conditions on the data and the source term of the problem in order to establish a foundation for regularization that is independent of the angle of the semigroup. If we then define, for example $-A$ to be the Laplace operator on some function space, we may retrieve applications such as identifying a source term within a backward diffusion problem.
In 2005, Horvath and Kiss [Proc. AMS] showed that \begin{equation}\frac{\lambda _{n}}{\lambda _{m}}\leq\frac{n^{2}}{m^{2}},~~n\gt m\
In the present paper, we give additional conditions on the single-barrier potential $q(x)$ for which Theorem 2.2 (in Horvath and Kiss [Proc. AMS]) and the estimate $(\ref{HKeq:1})$ remain valid. Namely, we prove that if $q(x)$ is a nonnegative and single-barrier potential (with transition point $x_{0}\in[0,1]$), such that $|q'(x)|\leq q^{*}$ where $q^{*}=\frac{2}{15}\min\{q(0),
Note that, our approach used in this paper can be applied to the case of nonnegative and single-well potentials studied in Horvath and Kiss [Proc. AMS] (without further restrictions on $q(x)$).
We discuss new eigenvalue enclosures for indefinite Sturm-Liouville operators on the real line.
Enss (1983) proved a propagation estimate for the usual free Schrödinger operator that turned out to be very useful for inverse scattering by Enss-Weder (1995). Since then, this method has been called the Enss-Weder time-dependent method. In this talk, first, we introduce the same type propagation estimate for the fractional powers of negative Laplacian. Second, we report about the inverse scattering by applying the Enss-Weder time-dependent method. We find that the high velocity limit of the scattering operator uniquely determines the short-range interactions.
The main objective of the talk is to illustrate how one can construct Schrödinger operators on a bounded interval with predefined essential and discrete spectra. The required structure of the spectrum is realized by a special choice of a sequence of point interactions of $\delta$ and/or $\delta'$ types. Our construction is inspired by celebrated paper [R. Hempel, L. Seco, B. Simon, J. Funct. Anal. 102 (1991), 448-483] and its sequel [R. Hempel, T. Kriecherbauer, P. Plankensteiner, Math. Nachr. 188 (1997), 141-168], where a similar problem was treated for Neumann Laplacians on bounded domains.
This is a joint work with Jussi Behrndt (TU Graz).
We consider Zakharov-Shabat systems of the form \[v_1'= -i\xi v_1 +q(t) v_2, \quad v_2'= -q(t) v_1+i\xi v_2 \] where $\xi$ is a complex spectral parameter and $q(t)$ is a real potential satisfying certain assumptions. In particular, $q(t)$ will be slowly decaying of the form $\sim |t|^{-\gamma}$, with $0 \lt \gamma \le 1$.
For such potentials, there may exist infinitely many purely imaginary eigenvalues $\xi_k=is_k$ ($k=1,2,\dots$) with $s_1\gt s_2\gt \dots \gt 0.$ For a given $s\gt 0,$ let $N(s)=\#\{j:s_j\gt s\}.$ The main part of this talk will be devoted to the problem of finding the leading asymptotics of $N(s)$ as $s \to 0$. This further leads to estimates for the moments of eigenvalues, that is, for sums of the form $\sum_{j} s_j^{\alpha}$ (for suitable $\alpha\gt 0$) in terms of integrals of powers of $q(t).$ The methods employed include the Prüfer transformation, an analog of Dirichlet-Neumann bracketing, and a rigorous WKB analysis.
One of the possible ways of formulating nonrelativistic quantum mechanics in the phase space is via the Wigner function. Its time evolution is given by the Moyal equation. Assuming the simple case of isolated quantum system, our first goal is to prove the existance of one-parameter unitary group describing the time evolution of the system. The generator of this group is a sum of two noncommuting symmetric operators and at least one of them has to be unbounded. To solve the Moyal equation numerically we used spectral split operator algorithm, based on the Kato-Trotter formula $$\exp [t(A+B)]=\exp(tA)\exp(tB)+O(t^2)$$ and the Strang formula $$\exp\left[t(A+B)\right]=\exp\left(\frac{tA}{2}\right)\exp(tB)\exp\left(\frac{tA}{2}\right)+O(t^3).$$
This approach is present in the literatrure in the case of Schrödinger equation, but so far it was not done for the Moyal equation. Furthermore, we plan to estimate not only the errors caused by the use of above approximations of the exponent $\exp[t(A+B)]$, but also the discretization error, which appears in numerical implementation of the algorithm.
Kołaczek, D. et al. (2020). Phase-space approach to time evolution of quantum states in confined systems. The spectral split-operator method, in P. Kulczycki, J. Kacprzyk, L. Koczy, R. Mesiar and R. Wisniewski (Eds), Information Technology, Systems Research, and Computational Physics, Springer, Cham, p. 307.
Neidhardt H. et al. (2018). Remarks on the Operator-Norm Convergence of the Trotter Product Formula, Integral Equations and Operator Theory 90(2):15.
Hansen, E. et al. (2009). Exponential splitting for unbounded operators, Mathematics of Computation 78(267):1485.
Spectrum of operators plays a very important role in the study of the associated physical systems. Computation of spectrum is a very difficult problem, especially when the operator under consideration acts on infinite dimensional spaces. Various kind of approximation techniques are used in such situation to get information about spectrum. The literature in this direction covers truncation method, perturbation techniques, approximation in the distributional sense, approximation with respect various metrics like Hausdorff metric on compact sets etc. The operator algebraic approach to this problem was initiated by Arveson. For a bounded self-adjoint operator $A$ on a separable Hilbert space $\mathbb{H}$, the usage of eigenvalue sequence of the finite dimensional truncations $A_n=P_nAP_n$, ($P_n$ is a sequence of orthogonal projections) to approximate the spectrum $\sigma(A)$ of $A$ is well known. In the recent literature, we tried to use this technique to continuous and random family of bounded operators respectively. A similar approach is followed by Ben-Artzi and Thomas Holding to a holomorphic family of unbounded operators. Now we consider the problem to extend this approximation techniques to a random family of unbounded operators and hence to obtain better estimates for the spectral distribution of Wigner operators. There are good estimates for the eigenvalue distribution of Wigner matrices. We wish to use them in this regard. In this talk, I also wish to present some of the spectral gap prediction results.
We provide numerous bounds for eigenvalues of the non-self-adjoint operator associated with the damped wave equation. Our approach is based on the Birman-Schwinger principle and Lieb-Thirring-type inequalities. This is joint work with David Krejcirik.
Assuming finite derivatives with arbitrary small rational steps of discretization, we show that non-linear stochastic ODEs, linear ODEs, and linear PDEs (all of them with bounded continuous and discontinuous coefficients) generate the same $C^*$-algebra, namely the universal UHF-algebra. Thus, topologically and algebraically, they are equivalent.
More precisely, 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}$.
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 ${\mathscr H}_{1,1}$. So, the multidimensional world can be emulated by the one-dimensional one.
The effects of non-linear and stochastic terms and extended algebras of integrodifferential operators are also discussed. Some of the results are published in
[1] Anton A. Kutsenko (2017) Mixed multidimensional integral operators with piecewise constant kernels and their representations, Linear and Multilinear Algebra, DOI: 10.1080/03081087.2017.1415294
In this talk, we will characterise the positive, self-adjoint extensions of the minimal operator associated to the general discrete Sturm-Liouville expression (a second order difference operator). To do this we follow the method of Brown and Evans for continuous Sturm-Liouville expressions, utilising results from Kreĭn-Vishik-Birman theory. In particular, we make use of the one-to-one correspondence between the positive, self-adjoint extensions of an operator $T$ and the set of positive, self-adjoint operators that act in the kernel of $T^*$, the adjoint of $T$, and a known decomposition of the corresponding sesquilinear form in terms of the form associated to the Friedrichs extension.
Given the abstract evolution equation \begin{equation*}y'(t)=Ay(t),\ t\in \mathbb{R},\end{equation*} with a scalar type spectral operator $A$ in a complex Banach space, found are conditions necessary and sufficient for all weak solutions of the equation, which a priori need not be strongly differentiable, to be strongly infinite differentiable or strongly Gevrey ultradifferentiable of order $\beta\ge 1$, in particular analytic or entire, on $\mathbb{R}$. Also, revealed are certain interesting inherent smoothness improvement effects. The important particular case of the equation with a normal operator $A$ in a complex Hilbert follows immediately.
We consider the following implicit linear non-homogeneous difference equation of the second order:
\begin{equation}cx_{n+2}=bx_{n+1}+ax_n-f_n,\quad n=0,1,2,...,\end{equation} where $a,b,c$ and $f_n$ are known integers, $a\not=0$, $c\not=0$, $c\not=\pm1$ and $b$ or $a$ is not divisible by $c$.
Theorem 1. Suppose $c$ and $b$ have a common prime divisor $p$, but $a$ is not divisible by $p$. Then Equation (1)
has a unique solution over the ring of integer $p$-adic numbers $\mathbb{Z}_p$ and this solution may be found as \begin{equation}\label{Cauchy}x_n =\sum_{k=0}^{\infty}y_{k+1}\frac{f_{n+k}}{a}, \quad n=0,1,2,...,\end{equation} where the sequence $\{y_n\}_{n=0}^{\infty}$ belongs to $\mathbb{Q}\cap\mathbb{Z}_p$ and is the solution of the following initial problem \[\left\{\begin{array}{ccc}ay_{n+2}=-by_{n+1}+cy_n,\quad n=0,1,2,...,\\ y_0=0,\ y_1=1\end{array}\right. .\] All terms of the series (2) belong to the ring $\mathbb{Z}_p$ and the series (2) converges in the topology of this ring.
Consider now Equation (1) as an infinite system of linear equations. Suppose the conditions of Theorem 1 are fulfilled. Since $a$ is not divisible by $p$, after dividing (1) by $a$ we obtain the following equivalent system over the ring of integer $p$-adic numbers $\mathbb{Z}_p$: \begin{equation}\frac{c}{a}x_{n+2}=\frac{b}{a}x_{n+1}+x_n-\frac{f_n}{a},\ n=0,1,2,...\end{equation} Let us view elements of the set $S(\mathbb{Z}_p)$ of all sequences $x=\{x_n\}_{n=0}^{\infty}$
from $\mathbb{Z}_p$ as column vectors. Rewrite (3) in the such operator-vector form $${\cal A}x=\frac{f}{a},\quad {\cal A}=\left(\begin{matrix}1 & ba^{-1} & -ca^{-1} & 0 &\cdots\\0 & 1 & ba^{-1} & -ca^{-1} &\cdots\\0 & 0 & 1 & ba^{-1} & \cdots\\ \vdots & \vdots & \vdots & \vdots &\ddots \end{matrix}\right),\ x\in S(\mathbb{Z}_p),$$ where elements of the column vector $f=\{f_n\}_{n=0}^{\infty}$ are integers. The following theorem shows that the unique solution over $\mathbb{Z}_p$ may be found, using an analog of Cramer's rule.
Let ${\cal A}_n$ be the matrix formed by replacing the $n$-th column of ${\cal A}$ by $\frac{f}{a}\ (n=0,1,2,...)$, i.e.
$${\cal A}_0=\left(\begin{matrix}f_0a^{-1} & ba^{-1} & -ca^{-1} & 0 &\cdots\\ f_1a^{-1} & 1 & ba^{-1} & -ca^{-1} &\cdots\\ f_2a^{-1} & 0 & 1 & ba^{-1} & \cdots\\ \vdots & \vdots & \vdots & \vdots &\ddots \end{matrix}\right),$$ $${\cal A}_1=\left(\begin{matrix}1 & f_0a^{-1} & -ca^{-1} & 0 &\cdots\\ 0 & f_1a^{-1} & ba^{-1} & -ca^{-1} &\cdots\\ 0 & f_2a^{-1} & 1 & ba^{-1} & \cdots\\ \vdots & \vdots & \vdots & \vdots &\ddots \end{matrix}\right),...\ .$$ Denote by $\Delta_m$ (respectively $\Delta_{n,m}$) the $(m+1)$th order leading principal minor of the matrix ${\cal A}$ (respectively ${\cal A}_n$), $m,n=0,1,2,...$.
Theorem 2. Assume the conditions of Theorem 1 are fulfilled. Then Equation (1) has a unique solution over $\mathbb{Z}_p$. This solution may be found, using the following Cramer's rule: $$x_n=\frac{\det {\cal A}_n}{\det {\cal A}},\quad n=0,1,2,...,$$ where the determinants of ${\cal A},\ {\cal A}_n$ are defined as the limits in $\mathbb{Z}_p$ of the leading principal minors of these matrices, i.e. $$ \det {\cal A}=\lim\limits _{m\to\infty}\Delta_m,\quad \det {\cal A}_n=\lim\limits_{m\to\infty}\Delta_{n,m}.$$ An analog of Theorem 2 for the first order implicit linear non-homogeneous difference equation was obtained in [1].
Joint work with V. V. Martseniuk, S. L. Gefter and A. L. Piven, National University, Kharkiv, Ukraine.
In this talk I will present a joint work with X. Cabré and J. Solà-Morales where we study periodic solutions of nonlinear equations for integro-differential and multiplier operators. In this work we investigate both the variational formulation and the Hamiltonian structure, showing applications such as the construction of fractional Delaunay cylinders by purely variational techniques, the characterization of nonlinearities for which there exists a layer solution, and the description of the Fourier coefficients of half-harmonic maps. The talk will be mainly focused on the description of periodic solutions of integro-differential equations with Allen-Cahn and Benjamin-Ono type nonlinearities.
In $L_2(\mathbb{R}^d;\mathbb{C}^n)$, we consider a matrix elliptic second order differential operator $B_\varepsilon >0$. Coefficients of the operator $B_\varepsilon$ are periodic with respect to some lattice in $\mathbb{R}^d$ and depend on $\mathbf{x}/\varepsilon$. We study the quantitative homogenization for the solutions of the hyperbolic system $\partial _t^2\mathbf{u}_\varepsilon =-B_\varepsilon\mathbf{u}_\varepsilon$. In operator terms, we are interested in approximations of the operators $\cos (tB_\varepsilon ^{1/2})$ and $B_\varepsilon ^{-1/2}\sin (tB_\varepsilon ^{1/2})$ in suitable operator norms. Approximations for the resolvent $B_\varepsilon ^{-1}$ are already obtained by T.~A.~Suslina. So, we rewrite hyperbolic equation as parabolic system for the vector with components $\mathbf{u}_\varepsilon $ and $\partial _t\mathbf{u}_\varepsilon$, and consider corresponding unitary group. For this group, we adopt the proof of the Trotter-Kato theorem by introduction of some correction term and derive hyperbolic results from elliptic ones. For more details, see arXiv:1904.02781.
The aim of this talk is to present several counterexamples in (unbounded) operator theory using matrices of unbounded operators. Among the counterexamples, we deal with the case of the triviality of domains of powers of densely defined operators. Other counterexamples are related to commutators and self-commutators.
In this talk, we discuss the spectra of Schr\"odinger operators on carbon nanotubes with impurities from the point of view of the theory of quantum graphs. In the case of periodic Schrödinger operators on carbon nanotubes without impurities, it is known that the spectrum consists of infinitely many spectral bands and the set of eigenvalues with infinite multiplicities. In this talk, we give a finite number of impurities expressed as the $\delta$ vertex conditions to the operator. As a result, we obtain additional eigenvalues embedded in the interior of the spectral bands. Furthermore, we have an estimate from below of the number of embedded eigenvalues in each spectral bands for a suitable strength of $\delta$ vertex conditions.
In this talk, we deal with symmetric impurities with respect to $xy$-plane and rotation. Due to the rotational symmetry, we obtain a unitary equivalence between our operator and the direct sum of a finite number of Schrödinger operators on the degenerate carbon nanotube. Furthermore, we utilize the space-symmetry on $xy$-plane and decompose those operators as the direct sum of the reduced operators on half size degenerate carbon nanotube with the Dirichlet and Neumann boundary condition. After those decomposition, we examine the estimate from below of the number of eigenvalues in the spectral gaps of each reduced operator. Finally, we show that those eigenvalues are embedded in the spectral bands of other reduced operators.
JB*-triples form a class of complex Banach spaces which includes, for example, the Cartan factor $B(H,K)$ of bounded linear operators from a complex Hilbert space $H$ to a complex Hilbert space $K$, $C^\ast$-algebras and spin triples.
A defining property of a JB*-triple is the Jordan triple identity which is expressed in terms of the so-called box operators on the JB*-triple. In this talk we present recent results showing that certain Hermitian projections can be characterised in terms of box operators. The talk aims to be self-contained.
We present new spectral enclosures for the non-real spectrum of a class of $2\times2$ block operator matrices with self-adjoint operators $A$ and $D$ on the diagonal and operators $B$ and $-B^*$ as off-diagonal entries. Our main result resembles Gershgorin's circle theorem.
We improve known perturbation results for self-adjoint operators in Hilbert spaces and use these to prove spectral enclosures for $J$-self-adjoint operator matrices and perturbation theorems for $J$-non-negative operators. The results are applied to singular indefinite Sturm-Liouville operators. Known bounds on the non-real spectrum of such operators are improved.
A potential supported by line in a quantum waveguide is considered. The theory of self-adjoint extensions of symmetric operators is used to construct the potential. It is assumed that there is a small gap in the line. Asymptotics of resonances (in the width of the gap) is obtained. Method of matching of asymptotic expansions for boundary value problems is used. Comparison with cases of other boundary condition is made. Resonances for several coupling windows are investigated.
The work was partially supported by Russian Science Foundation, grant 16-11-10330.
Joint work with Alexei Vorobiev (ITMO University).
We consider two-dimensional canonical systems \begin{equation*} y'(t)=zJH(t)y(t),\quad t\in[a,b) , \end{equation*} where $J=\big(\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}\big)$ , $z\in \mathbb{C}$, and where the Hamiltonian $H:[a,b)\to \mathbb{R}^{2\times 2}$ is a.e. positive semidefinite, locally integrable and satisfies $\int_a^b\operatorname{tr} H(t) dt=\infty$.
To each Hamiltonian $H$ one can construct its Weyl coefficient $q_H$, which is a Nevanlinna function, i.e. an analytic function mapping the upper half-plane into itself. It is due to L.de Branges that this assignment is essentially one-to-one. It can be challenging to relate properties of the Hamiltonian to properties of its Weyl coefficient.
We establish upper and lower bounds of the imaginary part of $q_H(ir)$ for $r>0$ in terms of the Hamiltonian $H$. For large $r$, these bounds depend only on the behaviour of $H$ locally at $a$. The result can be seen as a generalisation of a theorem by I.S.Kac to the non-diagonal case.
For a large class of Hamiltonians the upper and lower bound coincide up to a factor, which is bounded and bounded away from zero. Hence, the behaviour of $\operatorname{Im} q_H(ir)$ for large $r$ can, in many cases, be determined up to constant factors.
In the proof we follow an approach used by H.Winkler. Estimates for the power series coefficients of the fundamental solution of the system give rise to estimates of centres and radii of Weyl's nested discs.
The main focus of this talk is the following matrix Cauchy problem for the Dirac system on the interval $[0,1]$:
\[D'(x)+\left[\begin{array}{cc}
0 & \sigma_1(x)\\
\sigma_2(x) & 0
\end{array}
\right]
D(x)=i\mu\left[\begin{array}{cc}
1 & 0\\
0 &-1
\end{array}
\right]D(x),\quad D(0)=\left[\begin{array}{cc}
1 & 0\\
0 & 1
\end{array}\right],\] where $D=\{d_{ij}(x)\}_{i,j=1}^2$, $\mu\in\mathbb{C}$ is a spectral parameter, and $\sigma_j\in L_2[0,1]$, $j=1,2$. We propose a new approach for the study of asymptotic behaviour of its solutions as $\mu\to \infty$ and $|{\rm Im}\,\mu|\le c$. As an application, we obtain new, sharp asymptotic formulas for eigenfunctions of Sturm-Liouville operators with singular potentials:\begin{align*}
y''(x)+q(x)y(x)&=\lambda y(x) ,\quad x\in [0,1],\\
y(0)=y(1)&=0, \end{align*} where a potential $q(x)=\sigma'(x),$ and $\sigma\in L_2[0,1]$.
The talk is based on joint work with Alexander Gomilko, accepted for publications in Journal of Spectral Theory (arXiv:1808.09272).
The subject of this work concerns to the inverse scattering and inverse spectral problems. The inverse scattering problem can be formulated as follows: do the knowledge of the far ﬁeld pattern uniquely determines the unknown coeﬃcients of the diﬀerential operator? Saito’s formula and uniqueness result, as well as the reconstruction of singularities, are obtained for the scattering problems (see, for example, [1], [2]). The inverse spectral problem can be formulated as follows: do the Dirichlet eigenvalues and the derivatives of the normalised eigenfunctions at the boundary determine uniquely the coeﬃcients of the diﬀerential operator? In the case of the Schrdinger operators (see [3]) and for the magnetic Schrdinger operators (see [4]) the knowledge of the Dirichlet eigenvalues and the normal derivatives of the normalised eigenfunctions at the boundary uniquely determine unknown potentials. In the present work we show that the knowledge of the Dirichlet eigenvalues and derivatives up to the third order of the normalised eigenfunctions at the boundary uniquely determine the coeﬃcients of the second order perturbation of the biharmonic operator (see [5]).
In 1986 Robert Devaney has been suggested the conditions of a chaotic system. let $X$ be a topological space and $f$ be a countinuous map form $X$ onto $X,$ R. Devaney says that, to classify a dynamical system as chaotic, it must have the following properties:
In infinite dimensional Banach space $X$ the linear map $T$ is said to chaotic if satisfies the conditions II and III. The condition II is equivalent to the following definition:
Definition. The continuous linear operator $T$ in a separable Banachspace $X$ is said to be hypercyclic if there exists a vector $x \in X$ such that the set $\operatorname{orb}(T,x) = \{T^n x,\ n \in \mathbb{N}_0:=\mathbb{N} \cup \{ 0 \} \} $ is dense in $X$.
In this talk the spectral properties of hypercyclic operator $T$ are given.
The Fourier transform, which lies at the heart of the classical theory of harmonic analysis, is generated by the eigenfunction expansion of the Sturm-Liouville operator $-{d^2 \over dx^2}$. This naturally raises a question: is it possible to generalize the main facts of harmonic analysis to integral transforms of Sturm-Liouville type?
In this talk we introduce a novel unified framework for the construction of product formulas and convolutions associated with a general class of regular and singular Sturm-Liouville boundary value problems. This unified approach is based on the application of the Sturm-Liouville spectral theory to the study of the associated hyperbolic equation. As a by-product, an existence and uniqueness theorem for degenerate hyperbolic Cauchy problems with initial data at a parabolic line is established.
We will show that each Sturm-Liouville convolution gives rise to a Banach algebra structure in the space of finite Borel measures in which various probabilistic concepts and properties can be developed in analogy with the classical theory. Examples will be given, showing that many known convolution-type operators — such as the Hankel, Jacobi and Whittaker convolutions — can be constructed using this general approach.
This talk is based on joint work with Manuel Guerra and Semyon Yakubovich.
The famous (generalized) Hilbert matrix represents one of the very few examples of the Hankel matrix operator with explicitly solvable spectral problem. Its diagonalization was carried out by Rosenblum in 1958 and is by no means trivial. A possible approach, which can be applied in order to diagonalize the Hilbert matrix, relies on finding a commuting Jacobi operator whose spectral problem turns out to be explicitly solvable. Such method allows an abstract description and is sometimes called the commutator method.
First, we recall the commutator method and discuss some of its limitations if applied to Jacobi operators associated with classical families of hypergeometric orthogonal polynomials from the Askey scheme. Second, we show that the commutator method can be successfully applied to Jacobi operators of the Stieltjes-Carlitz polynomials and find new explicitly diagonalizable Hankel matrices. The talk is based on a joint work with P. Šťovíček.
In $L_2({\mathbb R}^d;{\mathbb C}^n)$, we consider a selfadjoint matrix strongly elliptic operator $A_\varepsilon$, $\varepsilon >0$, given by the differential expression $b({\mathbf D})^* g({\mathbf x}/\varepsilon)b({\mathbf D})$. Here $g({\mathbf x})$ is a periodic bounded and positive definite matrix-valued function, and $b({\mathbf D})$ is a first order differential operator. We study the behavior of the operators $\cos( t A_\varepsilon^{1/2})$ and $A_\varepsilon^{-1/2} \sin( t A_\varepsilon^{1/2})$, $t\in {\mathbb R}$, for small $\varepsilon$. It is proved that, as $\varepsilon \to 0$, these operators converge to $\cos( t A_0^{1/2})$ and $A_0^{-1/2} \sin( t A_0^{1/2})$, respectively, in the norm of operators acting from the Sobolev space $H^s({\mathbb R}^d;{\mathbb C}^n)$ (with a suitable $s$) to $L_2({\mathbb R}^d;{\mathbb C}^n)$. Here $A_0 = b({\mathbf D})^* g^0 b({\mathbf D})$ is the effective operator. We prove sharp-order error estimates and study the question about the sharpness of the result with respect to the norm type. The results are applied to study the behavior of the solution $u_\varepsilon({\mathbf x},t)$ of the Cauchy problem for hyperbolic equation $\partial^2_t u_\varepsilon({\mathbf x},t) = (A_\varepsilon u_\varepsilon)({\mathbf x},t)$. Applications to the nonstationary equations of acoustics and elasticity are given. The method is based on the scaling transformation, the Floquet-Bloch theory and the analytic perturbation theory.
It is well-known by works of several authors that the spectrum of the Neumann-Laplace operator may be non-discrete even in bounded domains, if the boundary of the domain has some irregularities. In the same direction, in a paper in 2008 with S.\,Nazarov we considered the Steklov spectral problem in a bounded domain $\Omega \subset \mathbb{R}^n$, $n \geq 2$, with a peak and showed that the spectrum may be discrete or continuous depending on the sharpness of the peak. Later, we proved that the spectrum of the Robin Laplacian in non-Lipschitz domains may be quite pathological since, in addition to countably many eigenvalues, the residual spectrum may cover the whole complex plain.
We have recently complemented this study in two papers, where we consider the spectral Robin-Laplace- and Steklov-problems in a bounded domain $\Omega$ with a peak and also in a family $\Omega_\varepsilon$ of domains blunted at the small distance $\varepsilon >0$ from the peak tip. The blunted domains are Lipschitz and the spectra of the corresponding problems on $\Omega_\varepsilon$ are discrete. We study the behaviour of the discrete spectra as $\varepsilon \to 0$ and their relations with the spectrum of case with $\Omega$. In particular we find various subfamilies of eigenvalues which behave in different ways (e.g. "blinking" and "stable" families")
and we describe a mechanism how the discrete spectra turn into the continuous one in this process.
The work is a co-operation with Sergei Nazarov (St. Petersburg) and also Nicolas Popoff (Bordeaux).
We will present a review of the spectral analysis on fractal spaces with the spectral decimation property. For such spaces symmetry allows to compute eigenvalues and eigenfunctions by recursive formulas involving rational functions. We will show how these techniques can be applied to solving the wave equation on fractals.
The reaction-diffusion problem on a growing domain cannot be solved analytically in general. However, the symmetries of the partial differential equations, which can be studied systematically, can be used to obtain certain types of analytic solutions, the so called invariant solutions. The invariant solutions can be used to gain some insight into large time behaviour of the solution and even to inspect instability.
A linear operator pencil is a first order polynomial with bounded operators as coefficients, $A(s) = sS_1 - T_1$, where $s$ is a complex number and $S_1$, $T_1$ are bounded operators. The essential spectrum of A is defined as the set of all $s$ such that $A(s)$ is no (semi-) Fredholm operator. We investigate the question which perturbations of the coefficients do not change the essential spectrum. For this, consider a second operator pencil $B(s) = sS_2 - T_2$, where $S_2$ and $T_2$ are bounded operators. If $S_1-S_2$ and $T_1-T_2$ are two compact operators, then obviously also the difference $A(s) - B(s) = s(S_1 - S_2) - (T_1 - T_2)$ is compact and, hence, the essential spectra of the pencils $A$ and $B$ coincide. But the essential spectrum of two operator pencils may coincide even if the difference of the coefficients is substantial. For example, let $M$ be a bounded and boundedly invertible operator. Then obviously $A(s):= sI - T$ and $B(s):= sM - TM = A(s)M$ have the same essential spectrum. We will present some sufficient conditions which ensure that the essential spectra of $A$ and $B$ coincide. This is done by exploiting a strong relation between an operator pencil and a specific linear subspace (linear relation).
The talk is based on a joint paper with Hannes, TU Ilmenau, Germany, Nedra Moalla, Sfax, Tunisia, Friedrich Philipp, KU Eichstätt, Germany, and Wafa Selmi, Sfax, Tunisia.
Let $A$ be a linear operator from a Banach space $\mathcal X$ to another Banach space $\mathcal Y$ and let $R(A)$ and $N(A)$ denote the range and null space of $A$ respectively. It is well known that the quantities $\alpha (A):= \dim N(A)$, $\beta (A):=\dim \mathcal Y/R(A)$ and the index $\kappa(A):=\alpha(A)-\beta(A)$ have some kind of stability when subjected to a small perturbation under certain conditions (see \cite{Kato}).
In this talk, we discuss the eigenvalue problem $\mathcal Tx \subset \lambda\mathcal A x$ where $\mathcal A$ and $\mathcal T$ are linear relations on a Hilbert space $\mathcal H$ and its relationship to the stability of index and related quantities for linear relations.
In this talk we provide a Hilbert space perspective to the Dirichlet-to-Neumann operator on general domains. The main technical trick to avoid issues with the regularity of the boundary is to use abstract boundary data spaces which are an abstract implementation of 1-harmonic functions. This perspective allows one to obtain convergence of a sequence of Dirichlet-to-Neumann operators induced by a sequence of variable coefficients. There is no regularity assumption on the sequence of coefficients needed and the convergence of these is strictly weaker than convergence in the strong operator topology. This assumed convergence is in concrete cases equivalent to $H$-convergence in the sense of Murat and Tartar.
The talk is based on joint work with A. F. M. ter Elst and G. Gorden and can be found in [Elst, A. F. M. ter; Gorden, G. and Waurick, M. The Dirichlet-to-Neumann operator for divergence form problems. Annali di Matematica Pura ed Applicata, 198(1): 177-203, 2019].
We discuss the spectral theory of the first-order system $Ju'+qu=wf$ of differential equations on the real interval $(a,b)$ when $J$ is a constant, invertible skew-Hermitian matrix and $q$ and $w$ are matrices whose entries are distributions of order zero with $q$ Hermitian and $w$ non-negative. We do not require the definiteness condition often made on the coefficients of the equation.
Specifically, we construct associated minimal and maximal relations, and study self-adjoint restrictions of the maximal relation. For these we construct Green's function and prove the existence of a spectral (or generalized Fourier) transformation.
This is joint work with Kevin Campbell, Ahmed Ghatasheh, and Minh Nguyen.
We study the essential spectrum of operator pencils associated with anisotropic Maxwell equations with conductivity on finitely connected unbounded domains. The main result is that the essential spectrum of the Maxwell pencil is the union of two sets: namely, the spectrum of a divergence form operator pencil, and the essential spectrum of the Maxwell pencil with constant coefficients. This is joint work with G. Alberti, M. Brown and M. Marletta.
A Nevanlinna function, often also called Herglotz function, is an analytic function in the open upper half-plane $\mathbb C^+$ which has nonnegative imaginary part. Nevanlinna functions occur in the spectral analysis of differential operators whenever H.Weyl’s nested disks method is applicable.
By de Branges’ inverse spectral theorem, to each Nevanlinna function $q$ there corresponds an essentially unique two-dimensional canonical system $y'(t)=zJH(t)y(t)$ on the half-line with positive semidefinite Hamiltonian $H$, such that $q$ is the Weyl-coefficient of this system.
It is a common meta-principle that the high-energy behaviour of $q$, i.e., its behaviour towards $+i\infty$, is related to the behaviour of the corresponding Hamiltonian $H$ towards $0$. In recent work of J.Eckhardt, A.Kostenko, and G.Teschl, one instance of this principle was established. Namely, the existence of the radial limit (equivalently, nontangential limit) $\lim_{z\to i\infty}q(iy)$ was characterised in terms of $H$. This result — and the proof method — generalises former work of Y. Kasahara on Krein strings.
We consider the “non-convergent” situation, and investigate the set of limit points of $q$ towards $+i\infty$. We show that the following three properties of a subset $L$ of the closure of $\mathbb C^+$ in the Riemann-sphere are equivalent:
The core of the proof is an explicit construction of a Hamiltonian whose Weyl coefficient has a prescribed set of limit points.
The proof method is to refine and further exploit Kasahara’s rescaling trick.
We consider a rigged space $H_+\subset H\subset H_-$, where the central space $H$ may be either a Hilbert or a Krein space. In the first case, this is the usual concept of a rigged Hilbert space, also known as Gelfand triple. For a Krein space, the indefinite inner product of $H$ is used to construct the duality between $H_+$ and $H_-$. We now look at unbounded operators $T$ acting on $H_-$ with domain densely contained in $H_+$. We define a notion of selfadjointness for $T$ with respect to the central space $H$ and study resulting properties of the spectrum and estimates for the resolvent of $T$. The results are applied to a block operator matrix of Hamiltonian type.
Although discrete symplectic systems naturally arise in the discrete calculus of variations as Jacobi systems obtained from the weak Pontryagin maximum principle and they provide also the proper discrete counterpart of linear Hamiltonian differential systems, their spectral theory was untouched for many years. In this talk we present our recent contributions to this theory in the setting of the (time-reversed) discrete symplectic system \[\label{Sla}\tag{S$_\lambda$}
z_{k}(\lambda)=\big(\mathcal S_{k}+\lambda\mathcal V_{k}\big)z_{k+1}(\lambda),\quad k\in\mathcal I_{\mathbb Z},\] where the discrete $\mathcal I_{\mathbb Z}$ is finite or unbounded from above, $\lambda\in\mathbb C$ is the spectral parameter, $\mathcal S_k$ and $\mathcal V_k$ are $2n\times 2n$ complex-valued matrices such that \begin{equation}\label{E:1.1}\mathcal S_k^*\mathcal J\mathcal S_k=\mathcal J,\quad \mathcal V_k^*\mathcal J\mathcal S_k\ \text{ is Hermitian},\quad \mathcal V_k^*\mathcal J\,\mathcal V_k=0
\end{equation} with the skew-symmetric $2n \times 2n$ matrix $\mathcal J:=\left(\begin{smallmatrix} 0 & I \\ -I & 0\end{smallmatrix}\right)$. In particular, we discuss the necessity of the approach based on linear relations and characterize all self-adjoint extensions of the minimal linear relation associated with ($\ref{Sla}$), see [1, 2, 3, 4].