Titchmarsh-Weyl M-function Research Articles

On Fourier expansions for systems of ordinary differential equations with distributional coefficients

We study the spectral theory for the first-order system Ju′+qu=wf of differential equations on the real interval (a,b) where J is a constant, invertible, skew-hermitian matrix and q and w are matrices whose entries are distributions of order 0 with q hermitian and w non-negative. Specifically, we construct a generalized Weyl-Titchmarsh m-function with corresponding spectral measure τ and a generalized Fourier transform after imposing certain conditions on J, q, and w. Different conditions are motivated and studied in the later sections. A Fatou-type identity needed for our result is recorded in the appendix.

A practical method for recovering Sturm–Liouville problems from the Weyl function

In the paper we propose a direct method for recovering the Sturm–Liouville potential from the Weyl–Titchmarsh m-function given on a countable set of points. We show that using the Fourier–Legendre series expansion of the transmutation operator integral kernel the problem reduces to an infinite linear system of equations, which is uniquely solvable if so is the original problem. The solution of this linear system allows one to reconstruct the characteristic determinant and hence to obtain the eigenvalues as its zeros and to compute the corresponding norming constants. As a result, the original inverse problem is transformed to an inverse problem with a given spectral density function, for which the direct method of solution from Kravchenko and Torba (2021 Inverse Problems 37 015015) is applied. The proposed method leads to an efficient numerical algorithm for solving a variety of inverse problems. In particular, the problems in which two spectra or some parts of three or more spectra are given, the problems in which the eigenvalues depend on a variable boundary parameter (including spectral parameter dependent boundary conditions), problems with a partially known potential and partial inverse problems on quantum graphs.

The Detectable Subspace for the Friedrichs Model: Applications of Toeplitz Operators and the Riesz\u2013Nevanlinna Factorisation Theorem

We discuss how much information on a Friedrichs model operator (a finite rank perturbation of the operator of multiplication by the independent variable) can be detected from ‘measurements on the boundary’. The framework of boundary triples is used to introduce the generalised Titchmarsh–Weyl M-function and the detectable subspaces which are associated with the part of the operator which is ‘accessible from boundary measurements’. In this paper, we choose functions arising as parameters in the Friedrichs model in certain Hardy classes. This allows us to determine the detectable subspace by using the canonical Riesz–Nevanlinna factorisation of the symbol of a related Toeplitz operator.

The Asymptotic Behavior of the Titchmarsh-Weyl m-function for a Dirac System on the Line

The Asymptotic Behavior of the Titchmarsh-Weyl m-function for a Dirac System on the Line

A Note on the Asymptotic and Threshold Behaviour of Discrete Eigenvalues inside the Spectral Gaps of the Difference Operator with a Periodic Potential

The asymptotic and threshold behaviour of the eigenvalues of a perturbed difference operator inside a spectral gap is investigated. In particular, applications of the Titchmarsh-Weyl m-function theory as well as the Birman-Schwinger principle is performed to investigate the existence and behaviour of the eigenvalues of the operator H0+λWn inside the spectral gap of H0 in the limits λ↑∞ and λ↓0.

The m-functions of discrete Schrödinger operators are sparse compared to those for Jacobi operators

We explore the sparsity of Weyl–Titchmarsh m-functions of discrete Schrödinger operators. Due to this, the set of their m-functions cannot be dense on the set of those for Jacobi operators. All this reveals why an inverse spectral theory for discrete Schrödinger operators via their spectral measures should be difficult. To obtain the result, de Branges theory of canonical systems is applied to work on them, instead of Weyl–Titchmarsh m-functions.

Scattering matrices and Dirichlet-to-Neumann maps

A general representation formula for the scattering matrix of a scattering system consisting of two self-adjoint operators in terms of an abstract operator valued Titchmarsh–Weyl m-function is proved. This result is applied to scattering problems for different self-adjoint realizations of Schrödinger operators on unbounded domains, Schrödinger operators with singular potentials supported on hypersurfaces, and orthogonal couplings of Schrödinger operators. In these applications the scattering matrix is expressed in an explicit form with the help of Dirichlet-to-Neumann maps.

Uniqueness from discrete data in an inverse spectral problem for a pencil of ordinary differential operators

We prove a pair of uniqueness theorems for an inverse problem for an ordinary differential operator pencil of second order. The uniqueness is achieved from a discrete set of data, namely, the values at the points −n2 (n ∈ N) of (a physically appropriate generalization of) the Weyl– Titchmarsh m-function m(λ) for the problem. As a corollary, we establish a uniqueness result for a physically motivated inverse problem inspired by Berry and Dennis (‘Boundary-conditionvarying circle billiards and gratings: the Dirichlet singularity’, J. Phys. A: Math. Theor. 41 (2008) 135203). To achieve these results, we prove a limit-circle analogue to the limit-point m-function interpolation result of Rybkin and Tuan (‘A new interpolation formula for the Titchmarsh– Weyl m-function’, Proc. Amer. Math. Soc. 137 (2009) 4177–4185); however, our proof, using a Mittag-Leffler series representation of m(λ), involves a rather different method from theirs, circumventing the A-amplitude representation of Simon (‘A new approach to inverse spectral theory, I. Fundamental formalism’, Ann. Math. (2) 150 (1999) 1029–1057). Uniqueness of the potential then follows by appeal to a Borg–Marˇcenko argument.

Dirichlet-to-Neumann maps, abstract Weyl–Titchmarsh M-functions, and a generalized index of unbounded meromorphic operator-valued functions

We introduce a generalized index for certain meromorphic, unbounded, operator-valued functions. The class of functions is chosen such that energy parameter dependent Dirichlet-to-Neumann maps associated to uniformly elliptic partial differential operators, particularly, non-self-adjoint Schrödinger operators, on bounded Lipschitz domains, and abstract operator-valued Weyl–Titchmarsh M-functions and Donoghue-type M-functions corresponding to closed extensions of symmetric operators belong to it.The principal purpose of this paper is to prove index formulas that relate the difference of the algebraic multiplicities of the discrete eigenvalues of Robin realizations of non-self-adjoint Schrödinger operators, and more abstract pairs of closed operators in Hilbert spaces with the generalized index of the corresponding energy dependent Dirichlet-to-Neumann maps and abstract Weyl–Titchmarsh M-functions, respectively.

The inverse resonance problem for left-definite Sturm–Liouville operators

We address inverse spectral and scattering problems for the half-line, left-definite Sturm–Liouville equation, −u″+qu=λwu. These problems have been considered recently as they are critical in integrating the Camassa–Holm equation. Previous results required that the support of w be free of gaps, relatively open intervals on which w=0 almost everywhere. We relax this condition and prove an inverse spectral theorem that tells to what extent the Weyl–Titchmarsh m-function or the spectral measure determines the coefficients of the equation. Note that, unlike the Schrödinger equation, knowing the spectral measure is not the same as knowing the m-function. We also prove an inverse resonance theorem that explains to what extent the eigenvalues and resonances determine the spectral measure (and, thus, the coefficients). Again, unlike the Schrödinger case, these data are not sufficient; the presence of w multiplying the spectral parameter complicates the analysis. However, we show that, in most cases, only one other number is needed to fully recover the spectral measure.

Spectral analysis of indefinite Sturm-Liouville operators

For a class of indefinite J-nonnegative Sturm-Liouville operators, we present a criterion of similarity to a self-adjoint operator. This criterion is formulated in terms of Weyl-Titchmarsh m-functions. Moreover, using this result, we obtain a criterion, as well as simple sufficient conditions, formulated in terms of the coefficients of a given Sturm-Liouville operator.

Asymptotic generalized value distribution of solutions of the Schrödinger equation

We develop the theory of generalized value distribution for the class of functions defined as boundary values of Herglotz functions. One of the main results is an estimate of asymptotic generalized value distribution for Herglotz functions translated by an increment iδ off the real axis. We also present precise relations between the employed measures, and the link with compositions of Herglotz functions. A case of particular interest which we study is that of the Weyl-Titchmarsh m-function associated with Sturm-Liouville differential operators. In this context, this generalized theory allows for the possibility to describe the asymptotic value distribution of solutions of the Schrodinger equation in terms of other measures than Lebesgue measure.

The similarity problem for indefinite Sturm–Liouville operators and the HELP inequality

We study two problems. The first one is the similarity problem for the indefinite Sturm–Liouville operator A=−(sgnx)dwdxdrdx acting in Lw2(−b,b). It is assumed that w,r∈Lloc1(−b,b) are even and positive a.e. on (−b,b).The second object is the so-called HELP inequality (∫0b1r̃|f′|dx)2≤K2∫0b|f|2w̃dx∫0b|1w̃(1r̃f′)′|2w̃dx, where the coefficients w̃,r̃∈Lloc1[0,b) are positive a.e. on (0,b).Both problems are well understood when the corresponding Sturm–Liouville differential expression is regular. The main objective of the present paper is to give criteria for both the validity of the HELP inequality and the similarity to a self-adjoint operator in the singular case. Namely, we establish new criteria formulated in terms of the behavior of the corresponding Weyl–Titchmarsh m-functions at 0 and at ∞. As a biproduct of this result we show that both problems are closely connected. Namely, the operator A is similar to a self-adjoint one precisely if the HELP inequality with w̃=r and r̃=w is valid.Next we characterize the behavior of m-functions in terms of coefficients and then these results enable us to reformulate the obtained criteria in terms of coefficients. Finally, we apply these results for the study of the two-way diffusion equation, also known as the time-independent Fokker–Planck equation.

The inverse scattering problem at fixed energy based on the Marchenko equation for an auxiliary Sturm–Liouville operator

A new approach is proposed to the solution of the quantum mechanical inverse scattering problem at a fixed energy. The method relates the fixed energy phase shifts to those arising in an auxiliary Sturm–Liouville problem via the interpolation theory of the Weyl–Titchmarsh m-function. Then a Marchenko equation is solved to obtain the potential. As an illustration test results are shown including the reconstructions of the piecewise constant and Coulomb-type potentials. A byproduct of the new framework is a demonstration of the connection found between the Gelfand–Levitan and Marchenko formalisms.

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,

On the determinant formula in the inverse scattering procedure with a partially known steplike potential

We are concerned with the inverse scattering problem for the full line Schrödinger operator −∂2x + q(x) with a steplike potential q a priori known on . Assuming is known and short range, we show that the unknown part of q can be recovered bywhere is the classical Marchenko operator associated with and is a trace class integral Hankel operator. The kernel of is explicitly constructed in terms of the difference of two suitably defined reflection coefficients. Since is not assumed to have any pattern of behavior at −∞, defining and analyzing scattering quantities becomes a serious issue. Our analysis is based upon some subtle properties of the Titchmarsh–Weyl m-function associated with .

The Hirota τ-function and well-posedness of the KdV equation with an arbitrary step-like initial profile decaying on the right half line

We are concerned with the Cauchy problem for the KdV equation on the whole line with an initial profile V0 which is decaying sufficiently fast at +∞ and arbitrarily enough (i.e. no decay or pattern of behaviour) at −∞. We show that this system is completely integrable in a very strong sense. Namely, the solution V(x, t) admits the Hirota τ-function representation where is a Hankel integral operator constructed from certain scattering and spectral data suitably defined in terms of the Titchmarsh–Weyl m-functions associated with the two half-line Schrödinger operators corresponding to V0. We show that V(x, t) is real meromorphic with respect to x for any t > 0. We also show that under a very mild additional condition on V0 representation (0.1) implies a strong well-posedness of the KdV equation with such V0's. Among others, our approach yields some relevant results due to Cohen, Kappeler, Khruslov, Kotlyarov, Venakides, Zhang and others.

On the singular Weyl–Titchmarsh function of perturbed spherical Schrödinger operators

We investigate the singular Weyl–Titchmarsh m-function of perturbed spherical Schrödinger operators (also known as Bessel operators) under the assumption that the perturbation q ( x ) satisfies x q ( x ) ∈ L 1 ( 0 , 1 ) . We show existence plus detailed properties of a fundamental system of solutions which are entire with respect to the energy parameter. Based on this we show that the singular m-function belongs to the generalized Nevanlinna class and connect our results with the theory of super singular perturbations.

Meromorphic solutions to the KdV equation with non-decaying initial data supported on a left half line

We show that under the Korteweg–de Vries flow an initial profile supported on (−∞, 0) from a very broad function class (without any decay assumption) instantaneously evolves into a meromorphic function with no poles on the real line. Our treatment is based on a suitable modification of the inverse scattering transform and a detailed investigation of the Titchmarsh–Weyl m-function. As a by-product, we improve some related results of others.

On the Marchenko inverse scattering procedure with partial information on the potential

For the Schrödinger operator H = −∂2x + q(x) on L2(−∞, ∞) with a real potential q(x) from a broad class of non-decaying functions, we extend the well-known Marchenko inverse scattering method to recover q(x) from a suitably defined reflection coefficient R and the knowledge of q(x) on (0, ∞). Our main tool is the Titchmarsh–Weyl m-function associated with the Dirichlet boundary condition at x = 0.