Spectral theory for pairs of differential operators

Bennewitz, Christer

doi:10.1007/bf02386032

Cited by 21 publications

(31 citation statements)

References 13 publications

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“…Let T 0 = T * 1 be the closure of T c . Theorem 1.4 in [4] established a simple generalization of the von Neumann formula for symmetric operators, i.e., T 1 = T 0 D λ D λ as a direct sum, for any non-real λ.…”

Section: Proposition 24mentioning

confidence: 98%

“…Later many authors have dealt with more or less general left-definite problems. In particular we mention a series of papers by Niessen, Schneider, and their collaborators on singular left-definite socalled S-hermitian systems; see, e.g., [28], and Bennewitz [4] and the references cited there. For a more recent contribution, see Kong, Wu and Zettl [25].…”

Section: Introductionmentioning

confidence: 99%

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Scattering and inverse scattering for a left-definite Sturm–Liouville problem

Bennewitz

Brown

Weikard

2012

Journal of Differential Equations

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This work develops a scattering and an inverse scattering theory for the Sturm-Liouville equation −u + qu = λwu where w may change sign but q 0. Thus the left-hand side of the equation gives rise to a positive quadratic form and one is led to a leftdefinite spectral problem. The crucial ingredient of the approach is a generalized transform built on the Jost solutions of the problem and hence termed the Jost transform and the associated PaleyWiener theorem linking growth properties of transforms with support properties of functions. One motivation for this investigation comes from the CamassaHolm equation for which the solution of the Cauchy problem can be achieved by the inverse scattering transform for −u + 1 4 u = λwu.

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Section: Proposition 24mentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Scattering and inverse scattering for a left-definite Sturm–Liouville problem

Bennewitz

Brown

Weikard

2012

Journal of Differential Equations

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“…and develop a spectral theory for (2.1) in this space (see [2], [3], [6] and others). This case is often referred to as the left definite case in contrast to the more standard right definite case where w ≥ 0.…”

Section: The Equationmentioning

confidence: 99%

A uniqueness result for one‐dimensional inverse scattering

Bennewitz¹,

Brown²,

Weikard³

2012

Mathematische Nachrichten

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Key words Inverse scattering, m-function, one-dimensional problems, left and right definite problems MSC (2010) 34K29We consider the whole-line inverse scattering problem for Sturm-Liouville equations which have constant coefficients on a half-line. Since in this case the reflection coefficient determines a Weyl-Titchmarsh m-function, it determines the coefficients up to some simple Liouville transformations. Given inverse spectral theory, proofs are fairly simple but provide extensions of known results as we require less smoothness and less decay than is customary.

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“…Consequently, the defect numbers of T min , see (4.7), can be written as 8) so that, in particular, by the von Neumann decomposition in Theorem 4.3…”

Section: Defect Numbers Of the Minimal Relationmentioning

confidence: 99%

Square‐integrable solutions and Weyl functions for singular canonical systems

Behrndt

Hassi

Snoo

et al. 2011

Mathematische Nachrichten

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Boundary value problems for singular canonical systems of differential equations of the formare studied in the associated Hilbert space L 2 Δ (ı). With the help of a monotonicity principle for matrix functions their square-integrable solutions are specified. This yields a direct treatment of defect numbers of the minimal relation and simultaneously makes it possible to assign certain boundary values to the elements of the maximal relation induced by the system of differential equations in L 2 Δ (ı). The investigation of boundary value problems for these systems and their spectral theory can be carried out by means of abstract boundary triplet techniques. This paper makes explicit the construction and the properties of boundary triplets and Weyl functions for singular canonical systems. Furthermore, the Weyl functions are shown to have a property similar to that of the classical Titchmarsh-Weyl coefficients for singular Sturm-Liouville operators: they single out the square-integrable solutions of the homogeneous systems of canonical differential equations.

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Spectral theory for pairs of differential operators

Cited by 21 publications

References 13 publications

Scattering and inverse scattering for a left-definite Sturm–Liouville problem

Scattering and inverse scattering for a left-definite Sturm–Liouville problem

A uniqueness result for one‐dimensional inverse scattering

Square‐integrable solutions and Weyl functions for singular canonical systems

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