The topology of isospectral manifolds of tridiagonal matrices

Tomei, Carlos

doi:10.1215/s0012-7094-84-05144-5

Cited by 88 publications

(102 citation statements)

References 4 publications

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“…The inverse problem for such operators is open, that is, there are no effective conditions on a spectral measure dρ that tell us that its associated Jacobi matrix has all a n = 1. (The isospectral manifold of general Jacobi matrices with a n ∈ R is discussed in [70]. )…”

Section: §5 Reconstruction Of a Finite Jacobi Matrix From Two Spectramentioning

confidence: 99%

m-Functions and inverse spectral analysis for finite and semi-infinite Jacobi matrices

1997

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Abstract. We study inverse spectral analysis for finite and semi-infinite Jacobi matrices H. Our results include a new proof of the central result of the inverse theory (that the spectral measure determines H). We prove an extension of Hochstadt's theorem (who proved the result in the case n = N ) that n eigenvalues of an N × N Jacobi matrix, H, can replace the first n matrix elements in determining H uniquely. We completely solve the inverse problemThere is an enormous literature on inverse spectral problems for − [1,29,[56][57][58][59][60]64] and references therein), but considerably less for their discrete analog, the infinite and semi-infinite Jacobi matrices (see, e.g., [3, 4, 6-8, 13-22, 24, 26-28, 30, 32, 37, 38, 42-44, 50-52, 61-63, 66, 67, 69-71]) and even less for finite Jacobi matrices (where references include, e.g., [9,10,23,25,39,40,41,[45][46][47][48]). Our goal in this paper is to study the last two problems using one of the most powerful tools from the spectral theory of − d 2 dx 2 + V (x), the m-functions of Weyl. Explicitly, we will study finite N × N matrices of the form:

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Section: §5 Reconstruction Of a Finite Jacobi Matrix From Two Spectramentioning

confidence: 99%

m-Functions and inverse spectral analysis for finite and semi-infinite Jacobi matrices

1997

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“…The n-dimensional Tomei manifold consists of all (n + 1) × (n + 1) real symmetric tridiagonal matrices, with fixed simple spectrum λ 0 < λ 1 < · · · < λ n (the manifold is independent of the choice of simple spectrum). These manifolds were introduced by Tomei [1984] and further studied by Davis [1987]. An important result of Tomei is that these manifolds support a very natural cellular decomposition, which we now describe.…”

Section: Appendix: Tomei Manifoldsmentioning

confidence: 84%

Comparing seminorms on homology

Lafont¹,

Pittet²

2012

Pacific J. Math.

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We compare the l 1 -seminorm · 1 and the manifold seminorm · man on n-dimensional integral homology classes. Crowley and Löh showed that for any topological space X and any α ∈ H n (X; ‫,)ޚ‬ with n = 3, the equality α man = α 1 holds. We compute the simplicial volume of the 3-dimensional Tomei manifold and apply Gaifullin's desingularization to establish the existence of a constant δ 3 ≈ 0.0115416, with the property that for any X and any α ∈ H 3 (X; ‫,)ޚ‬ one has the inequality δ 3 α man ≤ α 1 ≤ α man .

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“…The first example of this kind was described in 1984 by Tomei in his paper [2], where the topology of the isospectral variety J n of Jacobi n × n-matrices (i.e. real three-diagonal symmetric matrices) …”

Section: Topology Of the Isospectral Variety Of Jacobi Matricesmentioning

confidence: 99%

Integrable systems and the topology of isospectral manifolds

Penskoi

2008

Theor Math Phys

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The topology of isospectral manifolds of tridiagonal matrices

Cited by 88 publications

References 4 publications

m-Functions and inverse spectral analysis for finite and semi-infinite Jacobi matrices

m-Functions and inverse spectral analysis for finite and semi-infinite Jacobi matrices

Comparing seminorms on homology

Integrable systems and the topology of isospectral manifolds

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