arXiv subjects

Raphael Pruckner

Publications and source records attributed to Raphael Pruckner.

9 recordsLinked to original sources

arXiv2025

Canonical systems whose Weyl coefficients have regularly varying asymptotics

For a two-dimensional canonical system $y'(t)=zJH(t)y(t)$ on an interval $(0,L)$ with $0<L\le\infty$ whose Hamiltonian $H$ is a.e.\ positive semidefinite, denote by $q_H$ its Weyl coefficient. De~Branges' inverse spectral theorem states that the assignment $H\mapsto q_H$ is a bijection between trace-normalised Hamiltonians and Nevanlinna functions. We prove that $q_H$ has an asymptotics towards $i\infty$ whose leading term is some (complex) multiple of a regularly varying function if and only if the primitive $M$ of $H$ is regularly or rapidly varying at $0$ and its off-diagonal entries do not oscillate too much. The leading term in the asymptotics of $q_H$ towards $i\infty$ is related to the behaviour of $M$ towards $0$ by explicit formulae. The speed of growth in absolute value depends only on the diagonal entries of $M$, while the argument of the leading coefficient corresponds to the relative size of the off-diagonal entries. Translated to the spectral measure $μ_H$ and the Hamiltonian $H$, this means that the diagonal of $H$ determines the growth of the symmetrised distribution function of $μ_H$, and the relative size and sign distribution of its off-diagonal is a measure for the asymmetry of $μ_H$. The results are applied to Sturm--Liouville equations, Krein strings and generalised indefinite strings to prove similar characterisations for the asymptotics of the corresponding Weyl coefficients.

math.SP↗

arXiv2023

Canonical systems whose Weyl coefficients have dominating real part

For a two-dimensional canonical system $y'(t)=zJH(t)y(t)$ on the half-line $(0,\infty)$ whose Hamiltonian $H$ is a.e. positive semi-definite, denote by $q_H$ its Weyl coefficient. De Branges' inverse spectral theorem states that the assignment $H\mapsto q_H$ is a bijection between Hamiltonians (suitably normalised) and Nevanlinna functions. The main result of the paper is a criterion when the singular integral of the spectral measure, i.e. Re $q_H(iy)$, dominates its Poisson integral Im $q_H(iy)$ for $y\to+\infty$. Two equivalent conditions characterising this situation are provided. The first one is analytic in nature, very simple, and explicit in terms of the primitive $M$ of $H$. It merely depends on the relative size of the off-diagonal entries of $M$ compared with the diagonal entries. The second condition is of geometric nature and technically more complicated, but explicit in terms of $H$ itself. It involves the relative size of the off-diagonal entries of $H$, a measurement for oscillations of the diagonal of $H$, and a condition on the speed and smoothness of the rotation of $H$.

math.SP↗

arXiv2023

An upper bound for the Nevanlinna matrix of an indeterminate moment sequence

The solutions of an indeterminate Hamburger moment problem can be parameterised using the Nevanlinna matrix of the problem. The entries of this matrix are entire functions of minimal exponential type, and any growth less than that can occur. An indeterminate moment problem can be considered as a canonical system in limit circle case by rewriting the three-term recurrence of the problem to a first order vector-valued recurrence. We give a bound for the growth of the Nevanlinna matrix in terms of the parameters of this canonical system. In most situations this bound can be evaluated explicitly. It is sharp in the sense that for well-behaved parameters it coincides with the actual growth of the Nevanlinna matrix up to multiplicative constants.

math.CA↗

arXiv2023

Estimates for the Weyl coefficient of a two-dimensional canonical system

For a two-dimensional canonical system $y'(t)=zJH(t)y(t)$ on some interval $(a,b)$ whose Hamiltonian $H$ is a.e. positive semi-definite and which is regular at $a$ and in the limit point case at $b$, denote by $q_H$ its Weyl coefficient. De Branges' inverse spectral theorem states that the assignment $H\mapsto q_H$ is a bijection between Hamiltonians (suitably normalised) and Nevanlinna functions. We give upper and lower bounds for $|q_H(z)|$ and $\operatorname{Im} q_H(z)$ when $z$ tends to $i\infty$ non-tangentially. These bounds depend on the Hamiltonian $H$ near the left endpoint $a$ and determine $|q_H(z)|$ up to universal multiplicative constants. We obtain that the growth of $|q_H(z)|$ is independent of the off-diagonal entries of $H$ and depends monotonically on the diagonal entries in a natural way. The imaginary part is, in general, not fully determined by our bounds (in forthcoming work we shall prove that for "most" Hamiltonians also $\operatorname{Im} q_H(z)$ is fully determined). We translate the asymptotic behaviour of $q_H$ to the behaviour of the spectral measure $μ_H$ of $H$ by means of Abelian--Tauberian results and obtain conditions for membership of growth classes defined by weighted integrability condition (Kac classes) or by boundedness of tails at $\pm\infty$ w.r.t. a weight function. Moreover, we apply our results to Krein strings and Sturm--Liouville equations.

math-ph↗

arXiv2022

A growth estimate for the monodromy matrix of a canonical system

We investigate the spectrum of 2-dimensional canonical systems in the limit circle case. It is discrete and, by the Krein-de Branges formula, cannot be more dense than the integers. But in many cases it will be more sparse. The spectrum of a particular selfadjoint realisation coincides with the zeroes of one entry of the monodromy matrix of the system. Classical function theory thus establishes an immediate connection between the growth of the monodromy matrix and the distribution of the spectrum. We prove a generic and flexibel upper estimate for the monodromy matrix, use it to prove a bound for the case of a continuous Hamiltonian, and construct examples which show that this bound is sharp. The first two results run along the lines of earlier work of R.Romanov, but significantly improve upon these results. This is seen even on the rough scale of exponential order.

math.SP↗

arXiv2021

Limit behaviour of Weyl coefficients

We study the sets of radial or nontangential limit points towards $i\infty$ of a Nevanlinna function q. Given a nonempty, closed, and connected subset L of $C_+$ , we explicitly construct a Hamiltonian H such that the radial- and outer angular cluster sets towards $i\infty$ of the Weyl coefficient q H are both equal to L. Our method is based on a study of the continuous group action of rescaling operators on the set of all Hamiltonians.

math.FA↗

arXiv2018

Density of the spectrum of Jacobi matrices with power asymptotics

We consider Jacobi matrices $J$ whose parameters have the power asymptotics $ρ_n=n^{β_1} \left( x_0 + \frac{x_1}{n} + {\rm O}(n^{-1-ε})\right)$ and $q_n=n^{β_2} \left( y_0 + \frac{y_1}{n} + {\rm O}(n^{-1-ε})\right)$ for the off-diagonal and diagonal, respectively. We show that for $β_1 > β_2$, or $β_1=β_2$ and $2x_0 > |y_0|$, the matrix $J$ is in the limit circle case and the convergence exponent of its spectrum is $1/β_1$. Moreover, we obtain upper and lower bounds for the upper density of the spectrum. When the parameters of the matrix $J$ have a power asymptotic with one more term, we characterise the occurrence of the limit circle case completely (including the exceptional case $\lim_{n\to \infty} |q_n|\big/ ρ_n = 2$) and determine the convergence exponent in almost all cases.

math.SP↗

arXiv2015

Bounds on order of indeterminate moment sequences

We investigate the order $ρ$ of the four entire functions in the Nevanlinna matrix of an indeterminate Hamburger moment sequence. We give an upper estimate for $ρ$ which is explicit in terms of the parameters of the canonical system associated with the moment sequence via its three-term recurrence. Under a weak regularity assumption this estimate coincides with a lower estimate, and hence $ρ$ becomes computable. Dropping the regularity assumption leads to examples where upper and lower bounds do not coincide and differ from the order. In particular we provide examples for which the order is different from its lower estimate due to M.S.Livšic.

math.SP↗

arXiv2015

Functional Calculus for definitizable self-adjoint linear relations on Krein spaces

In the present note a functional calculus $ϕ\mapsto ϕ(A)$ for self-adjoint definitizable linear relation on Krein spaces is developed. This functional calculus is the proper analogue of $ϕ\mapsto \int ϕ\, dE$ in the Hilbert space situation. It also comprises the Spectral Theorem for self-adjoint definitizable operators on Krein spaces showing the existence of spectral projections.

math.FA↗