Applications of two parameter spectral theory to symmetric generalised eigenvalue problems

Binding, Paul; Browne, Patrick J.

doi:10.1080/00036818808839776

Cited by 31 publications

(48 citation statements)

References 18 publications

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“…nonsemisimple) eigenvalue for F n (/ij), with Hj-*n 0 as j-xx). Without loss of generality we may assume k } -*X Q by virtue of [5,Theorem 3.7]. Appealing to [13, Section VII.…”

Section: Corollary 22 //(*) Is Nondefective Then F and Hence H Ismentioning

confidence: 99%

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Multiparameter root vectors

Binding

1989

Proceedings of the Edinburgh Mathematical Society

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PreliminariesThe concept of "root vectors" is investigated for a class of multiparameter eigenvalue problems where W m (X) = T m -£S =1 X n V mn operate in Hilbert spaces H m and ieC k . Previous work on this "uniformly elliptic" class has demonstrated completeness of the decomposable tensors x t ® • • • ® x k in a subspace G of finite codimension in H = H 1^> - a r e constructed for the "root subspaces" corresponding to XeU k .

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“…nonsemisimple) eigenvalue for F n (/ij), with Hj-*n 0 as j-xx). Without loss of generality we may assume k } -*X Q by virtue of [5,Theorem 3.7]. Appealing to [13, Section VII.…”

Section: Corollary 22 //(*) Is Nondefective Then F and Hence H Ismentioning

confidence: 99%

“…In Section 3 we use analytic perturbation theory, cf. [5], to discuss the defective case by a limiting process, and we connect our work with (1.5) and (1.6). Section 4 is devoted to remarks on determination of the root vectors, on Jordan structure of the F n and on the case where one of the A n^0 .…”

Section: Introductionmentioning

confidence: 99%

Multiparameter root vectors

Binding

1989

Proceedings of the Edinburgh Mathematical Society

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“…[4]) shows that k = j+d+ whenever X¡ik > 9 so we may replace (4.4) by We observe in conclusion that X¡>k = Xj also depends on counting i via positive multiplicities d+ (X), and one must interpret the remarks following [2, Theorem 2.8] in this light. …”

Section: Elliptic Equations With Indefinite Weightsmentioning

confidence: 99%

A variational principle in Kreĭn space

Binding¹,

Najman²

1994

Trans. Amer. Math. Soc.

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Abstract. A variational characterization, involving a max-inf of the Rayleigh quotient, is established for certain eigenvalues of a wide class of definitizable selfadjoint operators Q in a Krein space. The operator Q may have continuous spectrum and nonreal and nonsemisimple eigenvalues; in particular it may have embedded eigenvalues. Various applications are given to selfadjoint linear and quadratic eigenvalue problems with weak definiteness assumptions.

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“…In the mathematical literature there is some discussion of these questions at an abstract [7] as well as at a more concrete [3] level.…”

Section: Level Crossing and Its Analogmentioning

confidence: 99%

Spectral inversion of an indefinite Sturm–Liouville problem due to Richardson

Shanley¹

2009

J. Phys. A: Math. Theor.

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Abstract. We study an indefinite Sturm-Liouville problem due to Richardson whose complicated eigenvalue dependence on a parameter has been a puzzle for decades. In atomic physics a process exists that inverts the usual Schrödinger situation of an energy eigenvalue depending on a coupling parameter into the socalled Sturmian problem where the coupling parameter becomes the eigenvalue which then depends on the energy. We observe that the Richardson equation is of the Sturmian type. This means that the Richardson and its related Schrödinger eigenvalue functions are inverses of each other and that the Richardson spectrum is therefore no longer a puzzle.

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Applications of two parameter spectral theory to symmetric generalised eigenvalue problems

Cited by 31 publications

References 18 publications

Multiparameter root vectors

Multiparameter root vectors

A variational principle in Kreĭn space

Spectral inversion of an indefinite Sturm–Liouville problem due to Richardson

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