Weyl-Titchmarsh Functions Research Articles

The local Borg-Marchenko uniqueness theorem for matrix-valued Schrödinger operators with locally smooth at the right endpoint potentials

We present a new expression for the Weyl-Titchmarsh matrix-valued function of a self-adjoint matrix-valued Schrödinger operator defined on the interval [ 0 , b ) , where 0 < b ≤ ∞ . Let H j = − d 2 d x 2 I m + Q j , j=1,2, be two self-adjoint Schrödinger operators in L 2 ( ( 0 , b ) ) m × m and Q 1 = Q 2 a.e. on the interval [ 0 , a ] , where a ∈ ( 0 , b ) . It is assumed that the potentials Q 1 and Q 2 are sufficiently smooth in the right neighborhood of the point a, where the right-hand derivatives of Q 1 = Q 2 at a coincide up to a certain order. Let M j ( z ) be the Weyl-Titchmarsh functions of H j = − d 2 d x 2 I m + Q j , j=1,2. As a specific application of this expression, we establish a high-energy asymptotic for the difference between M 1 ( z ) and M 2 ( z ) . Besides, new proofs are given for the local Borg-Marchenko uniqueness theorem and the high-energy asymptotics of the Weyl-Titchmarsh functions.

Discrete Self-adjoint Dirac Systems: Asymptotic Relations, Weyl Functions and Toeplitz Matrices

We consider discrete Dirac systems as an approach to the study of the corresponding block Toeplitz matrices, which in many ways completes the famous approach via Szegő recurrences and matrix orthogonal polynomials. We prove an analog of the Christoffel–Darboux formula and derive the asymptotic relations for the analog of reproducing kernel (using Weyl–Titchmarsh functions of discrete Dirac systems). These asymptotic relations are expressed also in terms of block Toeplitz matrices. We study the case of rational Weyl–Titchmarsh functions (and GBDT version of the Backlund–Darboux transformation of the trivial discrete Dirac system) as well. It is shown that block diagonal plus block semi-separable Toeplitz matrices (which are easily inverted) appear in this case.

On Realization of the Original Weyl–Titchmarsh Functions by Shrödinger L-systems

We study realizations generated by the original Weyl–Titchmarsh functions $$m_\infty (z)$$ and $$m_\alpha (z)$$ . It is shown that the Herglotz–Nevanlinna functions $$(-\,m_\infty (z))$$ and $$(1/m_\infty (z))$$ can be realized as the impedance functions of the corresponding Shrodinger L-systems sharing the same main dissipative operator. These L-systems are presented explicitly and related to Dirichlet and Neumann boundary problems. Similar results but related to the mixed boundary problems are derived for the Herglotz–Nevanlinna functions $$(-\,m_\alpha (z))$$ and $$(1/m_\alpha (z))$$ . We also obtain some additional properties of these realizations in the case when the minimal symmetric Shrodinger operator is non-negative. In addition to that we state and prove the uniqueness realization criteria for Shrodinger L-systems with equal boundary parameters. A condition for two Shrodinger L-systems to share the same main operator is established as well. Examples that illustrate the obtained results are presented in the end of the paper.

On the classes of explicit solutions of Dirac, dynamical Dirac and Dirac–Weyl systems with non-vanishing at infinity potentials, their properties and applications

We construct explicitly potentials, Darboux matrix functions and corresponding solutions of Dirac, dynamical Dirac and Dirac–Weyl systems using generalised Bäcklund-Darboux transformation (GBDT) in the important case of nontrivial initial systems. In this way, we construct explicit solutions of systems with non-vanishing at infinity potentials, including steplike and power-law growth potentials. Thus, the constructed potentials (systems) differ fundamentally from the actively studied case of GBDT for the trivial initial systems.Generalised matrix eigenvalues A (not necessarily diagonal) and corresponding generalised matrix eigenfunctions Π(x) of the nontrivial initial systems are used in the GBDT constructions in this paper. Explicit expressions for these Π(x) are new and the method of deriving these expressions may be applied to various other important problems.The case of Dirac–Weyl system, which is of interest in electron dynamics and graphene theory is studied in greater detail, and generalised separation of variables appears in our approach to the study of this system.Explicit expressions for Weyl–Titchmarsh functions (in the form of pseudo-realisations) are derived.

Direct Cauchy theorem and Fourier integral in Widom domains

We derive the Fourier integral associated with the complex Martin function in the Denjoy domain of the Widom type with the Direct Cauchy Theorem (DCT). As an application we study canonical systems and corresponding transfer matrices generated by reflectionless Weyl-Titchmarsh functions in such domains. The DCT property appears to be crucial in many aspects of the underlying theory.

Inverse Scattering at Fixed Energy on Three-Dimensional Asymptotically Hyperbolic Stäckel Manifolds

In this paper, we study an inverse scattering problem at fixed energy on three-dimensional asymptotically hyperbolic Stäckel manifolds having the topology of toric cylinders and satisfying the Robertson condition. On these manifolds the Helmholtz equation can be separated into a system of a radial ODE and two angular ODEs. We can thus decompose the full scattering operator into generalized harmonics and the resulting partial scattering matrices consist of a countable set of $2 \\times 2$ matrices whose coefficients are the so-called transmission and reflection coefficients. It is shown that the reflection coefficients are nothing but generalized Weyl–Titchmarsh functions associated with the radial ODE. Using a novel multivariable version of the Complex Angular Momentum method, we show that the knowledge of the scattering operator at a fixed non-zero energy is enough to determine uniquely the metric of the three-dimensional Stäckel manifold up to natural obstructions.

Dirac–Krein Systems on Star Graphs

We study the spectrum of a self-adjoint Dirac-Krein operator with potential on a compact star graph $\mathcal G$ with a finite number $n$ of edges. This operator is defined by a Dirac-Krein differential expression with summable matrix potentials on each edge, by self-adjoint boundary conditions at the outer vertices, and by a self-adjoint matching condition at the common central vertex of $\mathcal G$. Special attention is paid to Robin matching conditions with parameter $\tau \in\mathbb R\cup\{\infty\}$. Choosing the decoupled operator with Dirichlet condition at the central vertex as a reference operator, we derive Krein's resolvent formula, introduce corresponding Weyl-Titchmarsh functions, study the multiplicities, dependence on $\tau$, and interlacing properties of the eigenvalues, and prove a trace formula. Moreover, we show that, asymptotically for $R\to \infty$, the difference of the number of eigenvalues in the intervals $[0,R)$ and $[-R,0)$ deviates from some integer $\kappa_0$, which we call dislocation index, at most by $n+2$.

On a Weyl–Titchmarsh theory for discrete symplectic systems on a half line

Recently, Bohner and Sun [9] introduced basic elements of a Weyl–Titchmarsh theory into the study of discrete symplectic systems. We extend this development through the introduction of Weyl–Titchmarsh functions together with a preliminary study of their properties. A limit point criterion is described and characterized. Green’s function for the half-line is introduced as a limit of such functions in the regular case and half-line solutions obtained are seen to satisfy λ-dependent boundary conditions at infinity.

Spectral theory of elliptic operators in exterior domains

Diverse closed (and selfadjoint) realizations of elliptic differential expressions A = Σ0⩽|α|,|β|⩽m (−1) α D α a α,β (x)D β , a α,β (·) ∈ C ∞( $$ \bar \Omega $$ ) on smooth (bounded or unbounded) domains Ω in ℝ n with compact boundary ∂Ω are considered. Trace-ideal properties of powers of resolvent differences for these closed realizations of A are proved by using the concept of boundary triples and operator-valued Weyl-Titchmarsh functions, and estimates for negative eigenvalues of certain selfadjoint extensions of the nonnegative minimal operator are derived. Our results extend classical theorems due to Vishik, Povzner, Birman, and Grubb.

Weyl–Titchmarsh functions of vector-valued Sturm–Liouville operators on the unit interval

The matrix-valued Weyl–Titchmarsh functions M ( λ ) of vector-valued Sturm–Liouville operators on the unit interval with the Dirichlet boundary conditions are considered. The collection of the eigenvalues (i.e., poles of M ( λ ) ) and the residues of M ( λ ) is called the spectral data of the operator. The complete characterization of spectral data (or, equivalently, N × N Weyl–Titchmarsh functions) corresponding to N × N self-adjoint square-integrable matrix-valued potentials is given, if all N eigenvalues of the averaged potential are distinct.

Weyl–Titchmarsh Function of an Abstract Boundary Value Problem, Operator Colligations, and Linear Systems with Boundary Control

The paper defines the Weyl–Titchmarsh function for an abstract boundary value problem and shows that it coincides with the transfer function of some explicitly described linear boundary control system. On the ground of obtained results we explore interplay among boundary value problems, operator colligations, and the linear systems theory that suggests an approach to the study of boundary value problems based on the open systems theory founded in works of M. S. Livsic. Examples of boundary value problems for partial differential equations and calculations of their Weyl–Titchmarsh functions are offered as illustration. In particular, we give an independent derivation of the Weyl–Titchmarsh function for the three dimensional Schrodinger operator introduced by W.O. Amrein and D.B. Pearson. Relationships to the Schrodinger operator with singular potential supported by the unit sphere are clarified and other possible applications of the developed approach in mathematical physics are noted.

Weyl-Titchmarsh theory and Borg-Marchenko-type uniqueness results for CMV operators with matrix-valued Verblunsky coefficients

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 [19]) 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 matrixvalued Verblunsky coefficients as this is of independent interest and an essential ingredient in proving the corresponding Borg–Marchenko-type uniqueness theorems. Mathematics subject classification (2000): Primary 34E05, 34B20, 34L40; Secondary 34A55.

Weyl–Titchmarsh theory for CMV operators associated with orthogonal polynomials on the unit circle

We provide a detailed treatment of Weyl–Titchmarsh theory for half-lattice and full-lattice CMV operators and discuss their systems of orthonormal Laurent polynomials on the unit circle, spectral functions, variants of Weyl–Titchmarsh functions, and Green's functions. In particular, we discuss the corresponding spectral representations of half-lattice and full-lattice CMV operators.

On Krein's formula in indefinite metric spaces

In this paper we extend some of the recent results in connection with the Krein resolvent formula which provides a complete description of all canonical resolvents and utilizes Weyl–Titchmarsh functions in the spaces with indefinite metrics. We show that coefficients in Krein's formula can be expressed in terms of analogues of the von Neumann parametrization formulas in the indefinite case. We consider properties of Weyl–Titchmarsh functions and show that two Weyl–Titchmarsh functions corresponding to π-self-adjoint extensions of a densely defined π-symmetric operator are connected via linear-fractional transformation with the coefficients presented in terms of von Neumann's parameters.

On periodic matrix-valued Weyl–Titchmarsh functions

We consider a certain class of Herglotz–Nevanlinna matrix-valued functions which can be realized as the Weyl–Titchmarsh matrix-valued function of some symmetric operator and its self-adjoint extension. New properties of Weyl–Titchmarsh matrix-valued functions as well as a new version of the functional model for such realizations are presented. In the case of periodic Herglotz–Nevanlinna matrix-valued functions, we provide a complete characterization of their realizations in terms of the corresponding functional model. We also obtain properties of a symmetric operator and its self-adjoint extension which generate a periodic Weyl–Titchmarsh matrix-valued function. We study pairs of operators (a symmetric operator and its self-adjoint extension) with constant Weyl–Titchmarsh matrix-valued functions and establish connections between such pairs of operators and representations of the canonical commutation relations for unitary groups of operators in Weyl's form. As a consequence of such an approach, we obtain the Stone–von Neumann theorem for two unitary groups of operators satisfying the commutation relations as well as some extension and refinement of the classical functional model for generators of those groups. Our examples include multiplication operators in weighted spaces, first and second order differential operators, as well as the Schrödinger operator with linear potential and its perturbation by bounded periodic potential.

Dirac type and canonical systems: spectral and Weyl–Titchmarsh matrixfunctions, direct and inverse problems

The direct spectral problem for the matrix Dirac type systems with locallysummable potentials and the inverse spectral problem for the matrixDirac type systems with locally bounded potentials are solved on theinterval and on the half-line. A direct procedure to recover the potential bythe Weyl–Titchmarsh function is given also. Some important corollarieson the high-energy asymptotics of the Weyl–Titchmarsh functions andon the local uniqueness for the corresponding inverse problem follow.New results are obtained for the general type canonical systems also.