The essential spectrum of canonical systems

Remling, Christian; Scarbrough, Kyle

doi:10.1016/j.jat.2020.105395

Cited by 6 publications

(3 citation statements)

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“…In the case r ≡ 1 conditions (4.34), (4.35) were proved in [34]. Conditions similar to (4.34) can be found also in [15] for Sturm-Liouville operators and in [54], [52] for canonical systems.…”

Section: 2mentioning

confidence: 75%

Indefinite Sturm-Liouville operators in polar form

Ćurgus¹,

Derkach²,

Trunk³

2021

Preprint

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Let B + and B − be nonnegative symmetric operators with defect numbers (1, 1) in Hilbert spaces H + and H − . We consider the coupling A of operators A + = B + and A − = −B − which is self-adjoint, nonnegative and has a nonempty resolvent set in the Kreȋn space K with the fundamental decomposition K = H + [+]H − . Such an operator is definitizable and has at most two critical points at ∞ and 0. It is similar to a self-adjoint operator in a Hilbert space if and only if its critical points are regular and there are no Jordan chains at 0. We find criteria for the regularity of critical points of A formulated in terms of the abstract Weyl functions m + and m − of the operators B + and B − .A typical example of such coupling is the operator A generated in the Kreȋn space K = L 2 w (I) by the differential expression> 0 and (sgn t)w(t) > 0 a.e. on I. The operator A is self-adjoint in the Kreȋn space L 2 w (I) and can be considered as the coupling of the restrictions of A onto L 2 w − (I − ) and L 2 w + (I + ), where I − = (b − , 0), I + = (0, b + ), w − is the restriction of −w onto I − and w + is the restriction of w onto I + . For this operator the general criteria for the regularity of a critical point are reformulated in terms of the coefficients w and r.

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Section: 2mentioning

confidence: 75%

Indefinite Sturm-Liouville operators in polar form

Ćurgus¹,

Derkach²,

Trunk³

2021

Preprint

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“…However, by [Naimark and Everett, 1968], §24, Theorem 1, such results can be applied to both half lines separately to cover the full line case. For the transformation to a canonical system we proceed according to [Remling and Scarbrough, 2020], Section 5 -"An Example". The λ = 0 solutions from ( 35) are arranged in the fundamental matrix T 0 and Y as…”

Section: Justification Of Sfa Theory For the Singular Casementioning

confidence: 99%

Singular Sturm-Liouville Problems with Zero Potential (q=0) and Singular Slow Feature Analysis

Richthofer,

Wiskott

2020

Preprint

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A Sturm-Liouville problem (λwy = (ry ) + qy) is singular if its domain is unbounded or if r or w vanish at the boundary. Then it is difficult to tell whether profound results from regular Sturm-Liouville theory apply. Existing criteria are often difficult to apply, e.g. because they are formulated in terms of the solution function.We study the special case that the potential q is zero under Neumann boundary conditions and give simple and explicit criteria, solely in terms of the coefficient functions, to assess whether various properties of the regular case apply. Specifically, these properties are discreteness of the spectrum (BD), self-adjointness, oscillation (ith solution has i zeros) and that the ith eigenvalue equals the SFA delta value (the total energy) of the ith solution. We further prove that stationary points of each solution strictly interlace with its zeros (in singular or regular case, regardless of the boundary condition, for zero potential or if q < λw everywhere). If r w is bounded and of bounded variation, the criterion simplifies to requiring |w | w → ∞ at singular boundary points.This research is motivated by Slow Feature Analysis (SFA), a data processing algorithm that extracts the slowest uncorrelated signals from a high-dimensional input signal and has notable success in computer vision, computational neuroscience and blind source separation. From [Sprekeler et al., 2014] it is known that for an important class of scenarios (statistically independent input), an analytic formulation of SFA reduces to a Sturm-Liouville problem with zero potential and Neumann boundary conditions. So far, the mathematical SFA theory has only considered the regular case, except for a special case that is solved by Hermite Polynomials. This work generalizes SFA theory to the singular case, i.e. open-space scenarios.

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“…Removing the positivity assumption makes the corresponding considerations much more complicated. For instance, despite its fundamental importance, a discreteness criterion for 2 × 2 canonical systems was found only recently by R. Romanov and H. Woracek [30] (see also [29]). Surprisingly enough (at least to the authors), a discreteness criterion for indefinite strings (the case when the measure υ vanishes identically and ω is a real-valued Borel measure on [0, L)) has essentially been available since the 1970s, when C. A. Stuart [33] established a compactness criterion for integral operators in the Hilbert space L 2 [0, ∞) of the form J : f → ∞ 0 q(max( • , t))f (t)dt, (1.3) for a function q in L 2 loc [0, ∞).…”

Section: Introductionmentioning

confidence: 99%