An approach to universality using Weyl m-functions

Based on high energy expansions and Herglotz properties of Green and Weyl m-functions we develop a self-contained theory of trace formulas for Jacobi operators. In addition, we consider connections with inverse spectral theory, in particular uniqueness results. As an application we work out a new approach to the inverse spectral problem of a class of reflectionless operators producing explicit formulas for the coefficients in terms of minimal spectral data. Finally, trace formulas are applied to scattering theory with periodic backgrounds.

In the setting of adjoint pairs of operators we consider the question: to what extent does the Weyl M-function see the same singularities as the resolvent of a certain restriction A B of the maximal operator? We obtain results showing that it is possible to describe explicitly certain spaces $${\mathcal{S}}$$ and $$\tilde{\mathcal{S}}$$ such that the resolvent bordered by projections onto these subspaces is analytic everywhere that the M-function is analytic. We present three examples – one involving a Hain-Lüst type operator, one involving a perturbed Friedrichs operator and one involving a simple ordinary differential operators on a half line – which together indicate that the abstract results are probably best possible.

We consider semi-infinite Jacobi matrices with discrete spectrum. We prove that a Jacobi operator can be uniquely recovered from one spectrum and subsets of another spectrum and norming constants corresponding to the first spectrum. As a corollary, we obtain semi-infinite Jacobi analog of Marchenko's inverse spectral theorem for Schödinger operators, i.e. a Jacobi operator can be uniquely recovered from the Weyl m-function (or the spectral measure). We also solve our Borg–Marchenko-type problem under some conditions on two spectra, when missing part of the second spectrum and known norming constants have different index sets.

For the general one-dimensional Schrödinger operator −d2/dx2+q(x) with real q ∈ L1(ℝ), this paper presents a new series representation of the Jost solution which, in turn, implies a new asymptotic representation of the Weyl m-function for locally summable q. This representation is then applied to smooth potentials q to obtain Weyl m-function power asymptotics. The condition q(N) ∈ L1(x0, x0+δ), for N ∈ ℕ0, allows one to derive the (N+1) term for almost all x ∈ [x0, x0+δ), thereby refining a relevant result by Danielyan, Levitan and Simon.

We characterize the algebraic-geometric potentials for the Schrödinger and AKNS operators using the Weyl m-functions and the Floquet exponent for these operators. The characterization is this: among random ergodic Schrödinger operators, the alebraic-geometric potentials are those for which (i) the spectrum is a union of finitely many intervals (or bands); (ii) the Lyapounov exponent vanishes on the spectrum.

A real-space formulation of the Zak phase via Weyl m-functions

In this paper, we consider Sturm-Liouville problems on two symmetric disjoint intervals with two supplementary discontinuous conditions at an interior point. First, we investigate some spectral properties of boundary value problems, and obtain the asymptotic form of the eigenvalues and the eigenfunctions. Then, the eigenfunction expansion of Green’s function is presented and we prove the uniqueness theorems for the solution of the inverse problem, and reconstruct the Sturm-Liouville operator and the coefficients of boundary conditions using the Weyl m-function and spectral data. Also, numerical examples are presented.

On the missing eigenvalue problem for Dirac operators

The partial inverse spectral problem for Sturm–Liouville operators on a star-shaped graph was studied. The authors showed that if the potentials but one were known a priori, then the unknown potential on the whole interval can be uniquely determined by part of information of the potential and part of eigenvalues. The methods employed rest on the Weyl's m-function and theory concerning densities of zeros of entire functions.

Starting with an adjoint pair of operators, under suitable abstract versions of standard PDE hypotheses, we consider the Weyl M-function of extensions of the operators. The extensions are determined by abstract boundary conditions and we establish results on the relationship between the M-function as an analytic function of a spectral parameter and the spectrum of the extension. We also give an example where the M-function does not contain the whole spectral information of the resolvent, and show that the results can be applied to elliptic PDEs where the M-function corresponds to the Dirichlet to Neumann map.

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.

Physical observables may be derived from the Green function, the Scattering matrix and the Titchmarsh Weyl m-function. Such objects possess, at least for Schrödinger-type problems, compact spectral representations encompassing a set of complex poles. These, so-called non-redundant poles, are identical to the eigenvalues of the associated Schrödinger equation. The above-mentioned spectral representations allow identification of the individual contributions from these non-redundant poles to observable physical spectral features in terms of their respective residues. A classification of these poles as resonant and background-building ones is suggested. Model potential studies indicate that the expansions of the S-matrix, the Green function as well as the associated Titchmarsh-Weyl m-function, converge rapidly, making the method attractive from both conceptional and computational viewpoints. This presentation includes applications of the above method to problems in atomic and molecular physics as well as a discussion of the search for general and accurate numerical methods utilizing, in particular, the exterior complex scaling formulation.Key wordsS-matrixGreen functionTitchmarsh Weyl m-functionspectral representationresonant and background-building polesuniform and exterior complex scaling

The property that a Jacobi matrix is reflectionless is usually characterized either in terms of Weyl m-functions or the vanishing of the real part of the boundary values of the diagonal matrix elements of the resolvent. We introduce a characterization in terms of stationary scattering theory (the vanishing of the reflection coefficients) and prove that this characterization is equivalent to the usual ones. We also show that the new characterization is equivalent to the notion of being dynamically reflectionless, thus providing a short proof of an important result of [Breuer-Ryckman-Simon]. The motivation for the new characterization comes from recent studies of the non-equilibrium statistical mechanics of the electronic black box model and we elaborate on this connection. To appear in Commun. Math. Phys.

As is well known, the classical Titchmarsh–Weyl m-function for second order differential operators admits a generalisation to considerably larger classes of differential equations. In the applications, however, the use of the general M-matrix often turns out to be rather cumbersome. The paper interprets the m-function in terms of Hilbert space notions, and shows that, in a sense that is made precise, the classical m-function can be recovered as a part of the more complicated one. Applications of this trick lead to results on spectral multiplicity and on stability properties of the spectrum.

The Titchmarsh–Weyl theory is applied to the Schrödinger equation in the case when the asymptotic form of the solution is not known. It is assumed that the potential belongs to the Weyl’s limit-point classification. A rigorous analytical continuation of the Green’s function, obtained from the solution regular at the origin and the square integrable Weyl’s solution (regular at infinity), to the ‘‘unphysical’’ Riemann energy sheet is carried out. It is demonstrated how the Green’s function can be uniquely constructed from the Titchmarsh–Weyl m-function and its Nevanlinna representation. The behavior of the m-function in the neighborhood of poles is investigated. The m-function is decomposed in a, so called, generalized real part (Reg) and a generalized imaginary part (Img). Reg(m) is found to have a significant argument change upon pole passages. Img(m) is found to be a generalized spectral density. From the generalized spectral density, a spectral resolution of the differential operator and its resolvent is derived. In the expansion contributions are obtained from bound states, resonance states (Gamow states), and the ‘‘deformed continuum’’ given by the generalized spectral density. The present expansion theorem is applicable to the general partial differential operator via a decomposition into partial waves. The numerical procedure involves all quantum numbers l and m, but for convenience, and with the inverse problem in mind, this study is focused on the case when the rotational quantum number equals zero. The theory is tested numerically and analyzed for an analytic model potential exhibiting a barrier and decreasing exponentially at infinity. The potential is Weyl’s limit point at infinity and allows for an analytical continuation into a sector in the complex plane. An attractive feature of the generalized spectral density of the present potential is that the poles close to the real axis seem to exhaust or deflate the above-mentioned density inside the pole string. Outside this string the density rapidly approaches that of a free particle. This information is used to derive an approximate representation of the m-function in terms of poles and residues as well as free-particle background. In order to display the features mentioned above, the present study is accompanied with several plots of analytically continued quantities related to the Green’s function.