Weyl Titchmarsh Theory Research Articles

Weyl-Titchmarsh theory as applied to the singular non-sectorial Schrödinger operator. Conditions for discreteness of the spectrum and compactness of the resolvent

The spectral properties of the singular Schrödinger operator with complex-valued potential which takes values in a wider region than the half-plane, have been little studied. In general case, the operator is non-sectorial, and the numerical range coincides with the entire complex plane. In this situation we propose sufficient conditions for discreteness of the spectrum and compactness of the resolvent.

Square dA-integrable solutions of differential systems with measures on the semiaxis

This paper extends the Weyl-Titchmarsh theory to the case of generalized differential systems. It is that proved the characteristic matrix of the resolvent kernel of a differential system with measures belongs to a locus, which is a matrix analog of the Weyl disk. It is established that such matrix disks are nested and converge to the limiting disk or point depending on the limiting behavior of the radii. This limiting set plays an important role in discussing of the number of solutions to such system that are square dA-integrated on the semiaxis.

On the class of canonical systems corresponding to matrix string equations: general-type and explicit fundamental solutions and Weyl-Titchmarsh theory

Using linear similarity of a certain class of Volterra operators to the squared integration, we derive an important representation of the general-type fundamental solutions of the canonical systems corresponding to matrix string equations. Explicit fundamental solutions of such canonical systems are constructed (via the GBDT version of Darboux transformation) as well. Examples and applications to dynamical canonical systems are given. Explicit solutions of the dynamical canonical systems are constructed. Three appendices are dedicated to the Weyl-Titchmarsh theory for canonical systems, to the transformation of a subclass of canonical systems into matrix string equations (and of a smaller subclass of canonical systems into matrix Schrödinger equations), and to a linear similarity problem for Volterra operators.

On square integrable solutions of a fractional differential equation

In this paper we construct the Weyl–Titchmarsh theory for the fractional Sturm–Liouville equation. For this purpose we used the Caputo and Riemann–Liouville fractional operators having the order is between zero and one.

Singular Hamiltonian system with several spectral parameters

In this paper, the Weyl–Titchmarsh theory has been constructed for the singular 2n-dimensional (even order) Hamiltonian system with several spectral parameters. In particular, we consider that the left end point of the interval is regular and the right end point of the interval is singular for the Hamiltonian system with several parameters. Using the nested circles approach, we prove that at least n-linearly independent solutions are squarly integrable with respect to some matrix functions.

On square integrable solutions and principal and antiprincipal solutions for linear Hamiltonian systems

New results in the Weyl-Titchmarsh theory for linear Hamiltonian differential systems are derived by using principal and antiprincipal solutions at infinity. In particular, a non-limit circle case criterion is established and a close connection between the Weyl solution and the minimal principal solution at infinity is shown in the limit point case. In addition, the square integrability of the columns of the minimal principal solution at infinity is investigated. All results are obtained without any controllability assumption. Several illustrative examples are also provided.

Principal solution in Weyl-Titchmarsh theory for second order Sturm-Liouville equation on time scales

A connection between the oscillation theory and the Weyl--Titchmarsh theory for the second order Sturm--Liouville equation on time scales is established by using the principal solution. In particular, it is shown that the Weyl solution coincides with the principal solution in the limit point case, and consequently the square integrability of the Weyl solution is obtained. Moreover, both limit point and oscillatory criteria are derived in the case of real-valued coefficients, while a~generalization of the invariance of the limit circle case is proven for complex-valued coefficients. Several of these results are new even in the discrete time case. Finally, some illustrative examples are provided.

Real-Valued Algebro-Geometric Solutions of the Two-Component Camassa–Holm Hierarchy

We provide a construction of the two-component Camassa-Holm (CH-2) hierarchy employing a new zero-curvature formalism and identify and describe in detail the isospectral set associated to all real-valued, smooth, and bounded algebro-geometric solutions of the $n$th equation of the stationary CH-2 hierarchy as the real $n$-dimensional torus $\mathbb{T}^n$. We employ Dubrovin-type equations for auxiliary divisors and certain aspects of direct and inverse spectral theory for self-adjoint singular Hamiltonian systems. In particular, we employ Weyl-Titchmarsh theory for singular (canonical) Hamiltonian systems. While we focus primarily on the case of stationary algebro-geometric CH-2 solutions, we note that the time-dependent case subordinates to the stationary one with respect to isospectral torus questions.

Supersymmetry and Schrödinger-type operators with distributional matrix-valued potentials

Building on work on Miura's transformation by Kappeler, Perry, Shubin, and Topalov, we develop a detailed spectral theoretic treatment of Schrödinger operators with matrix-valued potentials, with special emphasis on distributional potential coefficients. Our principal method relies on a supersymmetric (factorization) formalism underlying Miura's transformation, which intimately connects the triple of operators (D, H_1, H_2) of the form D= \left(\begin{smallmatrix} 0 & A^* \\ A & 0 \end{smallmatrix}\right) \, \text{ in } \, L^2(\mathbb{R})^{2m} \, \text{ and } \, H_1 = A^* A, \;\; H_2 = A A^* \, \text{ in } L^2(\mathbb{R})^m. Here A= I_m (d/dx) + \phi in L^2(\mathbb{R})^m , with a matrix-valued coefficient \phi = \phi^* \in L^1_{\text{loc}}(\mathbb{R})^{m \times m} , m \in \mathbb{N} , thus explicitly permitting distributional potential coefficients V_j in H_j , j=1,2 , where H_j = - I_m \frac{d^2}{dx^2} + V_j(x), \quad V_j(x) = \phi(x)^2 + (-1)^{j} \phi'(x), \; j=1,2. Upon developing Weyl–Titchmarsh theory for these generalized Schrödinger operators H_j , with (possibly, distributional) matrix-valued potentials V_j , we provide some spectral theoretic applications, including a derivation of the corresponding spectral representations for H_j , j=1,2 . Finally, we derive a local Borg–Marchenko uniqueness theorem for H_j , j=1,2 , by employing the underlying supersymmetric structure and reducing it to the known local Borg–Marchenko uniqueness theorem for D .

Limit point and limit circle classification for symplectic systems on time scales

In this paper we study the limit point and limit circle classification for symplectic systems on time scales, which depend linearly on the spectral parameter. In a broader context, we develop a unified Weyl–Titchmarsh theory for continuous and discrete linear Hamiltonian and symplectic systems. Both separated and coupled boundary conditions are allowed. Our results include the study of the Weyl disks and circles and their limiting behavior, as well as a precise analysis of the number of linearly independent square integrable solutions. We also prove an analogue of the famous Weyl alternative. We connect and unify many known results in the Weyl–Titchmarsh theory for continuous, discrete, and special time scales systems and explain the differences between them. Some of our statements, in particular those connected with coupled endpoints or the Weyl alternative, are new even in the continuous time setting.

Weyl–Titchmarsh theory for discrete symplectic systems with general linear dependence on spectral parameter

In this paper we develop the Weyl–Titchmarsh theory for discrete symplectic systems with general linear dependence on the spectral parameter. We generalize and complete several recent results concerning these systems, which have the spectral parameter only in the second equation. Our new theory includes characterizations of the Weyl discs and Weyl circles, their limiting behaviour, properties of square summable solutions including the analysis of the exact number of linearly independent square summable solutions and limit point/circle criteria. Some illustrative examples are also provided.

On spectral theory for Schrödinger operators with operator-valued potentials

Given a complex, separable Hilbert space H, we consider differential expressions of the type τ=−(d2/dx2)+V(x), with x∈(a,∞) or x∈R. Here V denotes a bounded operator-valued potential V(⋅)∈B(H) such that V(⋅) is weakly measurable and the operator norm ‖V(⋅)‖B(H) is locally integrable.We consider self-adjoint half-line L2-realizations Hα in L2((a,∞);dx;H) associated with τ, assuming a to be a regular endpoint necessitating a boundary condition of the type sin(α)u′(a)+cos(α)u(a)=0, indexed by the self-adjoint operator α=α⁎∈B(H). In addition, we study self-adjoint full-line L2-realizations H of τ in L2(R;dx;H). In either case we treat in detail basic spectral theory associated with Hα and H, including Weyl–Titchmarsh theory, Greenʼs function structure, eigenfunction expansions, diagonalization, and a version of the spectral theorem.

Initial value problems and Weyl-Titchmarsh theory for Schrödinger operators with operator-valued potentials

We develop Weyl-Titchmarsh theory for self-adjoint Schrodinger operators Hα in L 2 ((a,b);dx;H ) associated with the operator-valued differential expression τ = −(d 2 /dx 2 ) + V(� ), with V : (a,b) → B(H ), and H a complex, separable Hilbert space. We assume reg- ularity of the left endpoint a and the limit point case at the right endpoint b. In addition, the bounded self-adjoint operator α = α ∗ ∈ B(H ) is used to parametrize the self-adjoint boundary condition at the left endpoint a of the type sin(α)u ' (a) +cos(α)u(a) = 0, with u lying in the domain of the underlying maximal operator Hmax in L 2 ((a,b);dx;H ) as- sociated with τ . More precisely, we establish the existence of the Weyl-Titchmarsh solution of Hα , the corresponding Weyl-Titchmarsh m-function mα and its Herglotz property, and deter- mine the structure of the Green's function of Hα . Developing Weyl-Titchmarsh theory requires control over certain (operator-valued) so- lutions of appropriate initial value problems. Thus, we consider existence and uniqueness of solutions of 2nd-order differential equations with the operator coefficient V , ( −y '' + (V −z)y = f on (a,b), y(x0) = h0, y ' (x0) = h1,

Weyl-Titchmarsh theory for Sturm-Liouville operators with distributional potentials

We systematically develop Weyl-Titchmarsh theory for singular differential operators on arbitrary intervals $(a,b) \subseteq \mathbb{R}$ associated with rather general differential expressions of the type \[ \tau f = \frac{1}{r} (- \big(p[f' + s f]\big)' + s p[f' + s f] + qf),] where the coefficients $p$, $q$, $r$, $s$ are real-valued and Lebesgue measurable on $(a,b)$, with $p\neq 0$, $r>0$ a.e.\ on $(a,b)$, and $p^{-1}$, $q$, $r$, $s \in L^1_{\text{loc}}((a,b); dx)$, and $f$ is supposed to satisfy [f \in AC_{\text{loc}}((a,b)), \; p[f' + s f] \in AC_{\text{loc}}((a,b)).] In particular, this setup implies that $\tau$ permits a distributional potential coefficient, including potentials in $H^{-1}_{\text{loc}}((a,b))$. We study maximal and minimal Sturm-Liouville operators, all self-adjoint restrictions of the maximal operator $T_{\text{max}}$, or equivalently, all self-adjoint extensions of the minimal operator $T_{\text{min}}$, all self-adjoint boundary conditions (separated and coupled ones), and describe the resolvent of any self-adjoint extension of $T_{\text{min}}$. In addition, we characterize the principal object of this paper, the singular Weyl-Titchmarsh-Kodaira $m$-function corresponding to any self-adjoint extension with separated boundary conditions and derive the corresponding spectral transformation, including a characterization of spectral multiplicities and minimal supports of standard subsets of the spectrum. We also deal with principal solutions and characterize the Friedrichs extension of $T_{\text{min}}$. Finally, in the special case where $\tau$ is regular, we characterize the Krein-von Neumann extension of $T_{\text{min}}$ and also characterize all boundary conditions that lead to positivity preserving, equivalently, improving, resolvents (and hence semigroups).

Uniqueness results for one-dimensional Schrödinger operators with purely discrete spectra

We provide an abstract framework for singular one-dimensional Schroedinger operators with purely discrete spectra to show when the spectrum plus norming constants determine such an operator completely. As an example we apply our findings to prove a new uniqueness results for perturbed quantum mechanical harmonic oscillators. In addition, we also show how to establish a Hochstadt-Liebermann type result for these operators. Our approach is based on the singular Weyl-Titchmarsh theory which is extended to cover the present situation.

New analytical option pricing models with Weyl–Titchmarsh theory

In this paper, we study the price of a European option written on an underlying asset price that follows general diffusion processes in the risk-neutral measure where r is the risk-free rate, q the...

Direct and inverse spectral theory of singular left-definite Sturm–Liouville operators

We discuss direct and inverse spectral theory of self-adjoint Sturm–Liouville relations with separate boundary conditions in the left-definite setting. In particular, we develop singular Weyl–Titchmarsh theory for these relations. Consequently, we apply de Brangesʼ subspace ordering theorem to obtain inverse uniqueness results for the associated spectral measure. The results can be applied to solve the inverse spectral problem associated with the Camassa–Holm equation.

Commutation methods for Schrödinger operators with strongly singular potentials

AbstractWe explore the connections between singular Weyl–Titchmarsh theory and the single and double commutation methods. In particular, we compute the singular Weyl function of the commuted operators in terms of the original operator. We apply the results to spherical Schrödinger operators (also known as Bessel operators). We also investigate the connections with the generalized Bäcklund–Darboux transformation.

Weyl-Titchmarsh Theory for Schrodinger Operators with Strongly Singular Potentials

We develop Weyl-Titchmarsh theory for Schroedinger operators with strongly singular potentials such as perturbed spherical Schroedinger operators (also known as Bessel operators). It is known that in such situations one can still define a corresponding singular Weyl m-function and it was recently shown that there is also an associated spectral transformation. Here we will give a general criterion when the singular Weyl function can be analytically extended to the upper half plane. We will derive an integral representation for this singular Weyl function and give a criterion when it is a generalized Nevanlinna function. Moreover, we will show how essential supports for the Lebesgue decomposition of the spectral measure can be obtained from the boundary behavior of the singular Weyl function. Finally, we will prove a local Borg-Marchenko type uniqueness result. Our criteria will in particular cover the aforementioned case of perturbed spherical Schroedinger operators.

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.