On the completeness of eigenvectors of right definite multiparameter problems

1984 ◽  
Vol 96 (1-2) ◽  
pp. 69-78 ◽  
Author(s):  
Hans Volkmer
Keyword(s):  
Eigenvalue Problem ◽  
Discrete Spectrum ◽  
Hilbert Spaces ◽  
Complete System ◽  
Expansion Theorem ◽  
Image Position ◽  
Adjoint Operators ◽  
The Given ◽  
Two Parameter

SynopsisWe prove that there exists a complete system of eigenvectors of the eigenvalue problemfor self-adjoint operators Tr and Vrs on separable Hilbert spaces Hr. It is assumed that(i) the operators Tr have discrete spectrum;(ii) the operators Vrs are bounded and commute for each r;(iii) the operators Vrs have the definite sign factor property.This theorem generalizes and improves a result of Cordes for two-parameter problems. The proof of the theorem depends on an approximation of the given eigenvalue problem by simpler problems, a technique which is related to Atkinson's proof of his expansion theorem.

Existence conditions for two-parameter eigenvalue problems

1981 ◽  
Vol 91 (1-2) ◽  
pp. 15-30 ◽  
Author(s):  
Paul Binding ◽  
Patrick J. Browne ◽  
Lawrence Turyn
Keyword(s):  
Hilbert Spaces ◽  
Eigenvalue Problems ◽  
Sufficient Conditions ◽  
Compact Resolvent ◽  
Existence Conditions ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Adjoint Operators ◽  
Bounded Below ◽  
Two Parameter

SynopsisWe discuss necessary and sufficient conditions for the existence of eigentuples λ=(λl,λ2) and eigenvectors x1≠0, x2≠0 for the problem Wr(λ)xr = 0, Wr(λ)≧0, (*), where Wr(λ)= Tr + λ1Vr2, r=1,2. Here Tr and Vrs are self-adjoint operators on separable Hilbert spaces Hr. We assume the Vrs to be bounded and the Tr bounded below with compact resolvent. Most of our conditions involve the conesWe obtain results under various conditions on the Tr, but the following is typical:THEOREM. If (*) has a solution for all choices ofT1, T2then (a)0∉ V1UV2,(b)V1∩(—V2) =∅ and (c) V1⊂V2∪{0}, V2⊈V1∪{0}. Conversely, if (a) and (b) hold andV1⊈V2∪∩{0}, V2⊈ then (*) has a solution for all choices ofT1, T2.

An abstract multiparameter spectral theory

1982 ◽  
Vol 92 (3-4) ◽  
pp. 193-204 ◽  
Author(s):  
P. A. Binding ◽  
A. Källström ◽  
B. D. Sleeman
Keyword(s):  
Eigenvalue Problem ◽  
Tensor Product ◽  
Spectral Theory ◽  
Hilbert Spaces ◽  
Product Space ◽  
Infinite Dimensions ◽  
Tensor Product Space ◽  
Image Position ◽  
Adjoint Operators

SynopsisWe consider the eigenvalue problemfor self-adjoint operators Ai and Bij on separable Hilbert spaces Hi. It is assumed that and Bij are bounded with compact. Various properties of the eigentuples λi, and xi are deduced under a “definiteness condition” weaker than those used by previous authors, at least in infinite dimensions. In particular, a Parseval relation and eigenvector expansion are derived in a suitably constructed tensor product space.

VI.—On Solvability of Some Two-Parameter Eigenvalue Problems in Hilbert Space

1968 ◽  
Vol 68 (1) ◽  
pp. 83-93
Author(s):  
Z. Bohte
Keyword(s):  
Hilbert Space ◽  
Power Series ◽  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Radius Of Convergence ◽  
Finite Radius ◽  
Image Position ◽  
Adjoint Operators ◽  
Two Parameter

SynopsisThis paper studies two particular cases of the general 2-parameter eigenvalue problem namelywhere A, B, B1, B2, C, C1, C2 are self-adjoint operators in Hilbert space, all except A being bounded. The disposable parameters λ and μ have to be determined so that the equations have non-trivial solutions x, y.On the assumption that the solution is known for ∊ = o, solutions are constructed in the form of series for λ, μ, x, y as power series in ∊ with finite radius of convergence.

X.—The Two Parameter Sturm-Liouville Problem for Ordinary Differential Equations

1971 ◽  
Vol 69 (2) ◽  
pp. 139-148
Author(s):  
B. D. Sleeman
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Liouville Problem ◽  
Sturm Liouville ◽  
Image Position ◽  
Two Parameter ◽  
General Conditions

SynopsisThis paper discusses the existence, under fairly general conditions, of solutions of the two-parameter eigenvalue problem denned by the differential equation,and three point Sturm-Liouville boundary conditions.

A Note on Klein’s Oscillation Theorem for Periodic Boundary Conditions

1975 ◽  
Vol 17 (5) ◽  
pp. 749-755 ◽  
Author(s):  
M. Faierman
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Oscillation Theory ◽  
Continuous Functions ◽  
Oscillation Theorem ◽  
Image Position ◽  
Two Parameter

Recently Howe [4] has considered the oscillation theory for the two-parameter eigenvalue problem1a1bsubjected to the boundary conditions2a2bwhere for i = 1, 2, — ∞<ai<bi<∞, and qi are real-valued, continuous functions in [ai, bi], pi is positive in [ai, biz], and pi(ai)=pi(bi). Furthermore, it is also assumed that (A1B2—A2B1)≠0 for all values of x1 and x2 in their respective intervals.

Existence conditions for eigenvalue problems generated by compact multiparameter operators

1984 ◽  
Vol 96 (3-4) ◽  
pp. 261-274 ◽  
Author(s):  
Paul Binding ◽  
Patrick J. Browne ◽  
Lawrence Turyn
Keyword(s):  
Hilbert Space ◽  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Separable Hilbert Space ◽  
Linear Operators ◽  
Compact Resolvent ◽  
Multiparameter Eigenvalue Problem ◽  
Existence Conditions ◽  
Image Position ◽  
Adjoint Operators

SynopsisLet T, V1,…, Vk denote compact symmetric linear operators on a separable Hilbert space H, and write W(λ) = T + λ1V1 + … + λkVk, λ = (λ1, …, λk) ϵ ℝk. We study conditions on the conerelated to solubility of the multiparameter eigenvalue problemwith W(λ)−I nonpositive definite. The main result is as follows.Theorem. If 0 ∉ V, then (*) is soluble for any T. If 0 ∈ V, then there exists T such that (*) is insoluble.We also deduce analogous results for problems involving self-adjoint operators with compact resolvent.

scholarly journals Elliptic multiparameter eigenvalue problems

1987 ◽  
Vol 30 (2) ◽  
pp. 215-228 ◽  
Author(s):  
P. A. Binding ◽  
K. Seddighi
Keyword(s):  
Hilbert Spaces ◽  
Eigenvalue Problems ◽  
Positive Definite ◽  
Ellipticity Condition ◽  
Image Position ◽  
Adjoint Operators ◽  
Bounded Below ◽  
Compact Resolvents

We study the eigenproblemwhereand Tm, Vmn are self-adjoint operators on separable Hilbert spaces Hm. We assume the Tm to be bounded below with compact resolvents, and the Vmn to be bounded and to satisfy an “ellipticity” condition. If k = 1 then ellipticity is automatic, and if each Tm is positive definite then the problem is “left definite”.

scholarly journals XXV.—The Generating Function of the Reciprocal of a Determinant

1905 ◽  
Vol 40 (3) ◽  
pp. 615-629
Author(s):  
Thomas Muir
Keyword(s):  
Generating Function ◽  
Specific Character ◽  
Partial Fraction ◽  
Homogeneous Functions ◽  
Series Form ◽  
General Proposition ◽  
In Series ◽  
Image Position ◽  
The Given

(1) This is a subject to which very little study has been directed. The first to enunciate any proposition regarding it was Jacobi; but the solitary result which he reached received no attention from mathematicians,—certainly no fruitful attention,—during seventy years following the publication of it.Jacobi was concerned with a problem regarding the partition of a fraction with composite denominator (u1 − t1) (u2 − t2) … into other fractions whose denominators are factors of the original, where u1, u2, … are linear homogeneous functions of one and the same set of variables. The specific character of the partition was only definable by viewing the given fraction (u1−t1)−1 (u2−t2)−1…as expanded in series form, it being required that each partial fraction should be the aggregate of a certain set of terms in this series. Of course the question of the order of the terms in each factor of the original denominator had to be attended to at the outset, since the expansion for (a1x+b1y+c1z−t)−1 is not the same as for (b1y+c1z+a1x−t)−1. Now one general proposition to which Jacobi was led in the course of this investigation was that the coefficient ofx1−1x2−1x3−1…in the expansion ofy1−1u2−1u3−1…, whereis |a1b2c3…|−1, provided that in energy case the first term of uris that containing xr.

On a singular eigenvalue problem

1968 ◽  
Vol 64 (2) ◽  
pp. 439-446 ◽  
Author(s):  
D. Naylor ◽  
S. C. R. Dennis
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Bessel Functions ◽  
Asymptotic Condition ◽  
Real Constant ◽  
Limit Circle ◽  
Expansion In Eigenfunctions ◽  
Image Position

Sears and Titchmarsh (1) have formulated an expansion in eigenfunctions which requires a knowledge of the s-zeros of the equationHere ka > 0 is supposed given and β is a real constant such that 0 ≤ β < π. The above equation is encountered when one seeks the eigenfunctions of the differential equationon the interval 0 < α ≤ r < ∞ subject to the condition of vanishing at r = α. Solutions of (2) are the Bessel functions J±is(kr) and every solution w of (2) is such that r−½w(r) belongs to L2 (α, ∞). Since the problem is of the limit circle type at infinity it is necessary to prescribe a suitable asymptotic condition there to make the eigenfunctions determinate. In the present instance this condition is

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