Zeta-determinants of Sturm–Liouville operators with quadratic potentials at infinity

Hartmann, Luiz; Lesch, Matthias; Vertman, Boris

doi:10.1016/j.jde.2016.11.033

Cited by 5 publications

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“…Other examples come from singular manifolds. The following example is based on the spectral properties of operators studied [20]. The Laplacian on a manifold with cuspidal singularities in the metric decomposes into an operator with discrete spectrum, and one with continuous spectrum.…”

Section: Acknowledgementsmentioning

confidence: 99%

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Dixmier traces and residues on weak operator ideals

Goffeng

Usachev

2020

Journal of Mathematical Analysis and Applications

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We develop the theory of modulated operators in general principal ideals of compact operators. For Laplacian modulated operators we establish Connes' trace formula in its local Euclidean model and a global version thereof. It expresses Dixmier traces in terms of the vector-valued Wodzicki residue. We demonstrate the applicability of our main results in the context of log-classical pseudo-differential operators, studied by Lesch, and a class of operators naturally appearing in noncommutative geometry.where Res W (A) is the Wodzicki residue of A and Tr ω is any Dixmier trace.This theorem reduces the computation of Dixmier traces to that of the Wodzicki residue, which in turn is defined from the principal symbol of an operator and is therefore more accessible to computations. The spectral asymptotics of positive pseudo-differential operators was well known prior to Connes' trace formula by means of a Weyl law, see for instance [22, Chapter 29.1], but Connes' trace formula conceptualized Dixmier traces as an "integral" in noncommutative geometry. There were several attempts to extend and generalise Connes' trace formula (see e.g. [3], [15] and [19] for the treatment of anisotropic pseudo-differential operators, the perturbed Laplacian on the noncommutative two tori and manifolds with boundaries, respectively). Recent developments.A recent extension of the above theorem appeared in [23] (see also [26]). It expresses the Dixmier traces of so-called Laplacian modulated operators (for rigorous definitions, see Section 4 below) in terms of a vector-valued Wodzicki residue, which again depends on the symbol of an operator. We state the main result of [23] here. Let ℓ ∞ (N) denote the Banach space of bounded sequences with supremum norm. Example 2.8. Let ϕ be a function satisfying (3) and (M, g) a closed Riemannian manifold. The Weyl law guarantees the eigenvalue asymptotics λ k ((1+∆ g ) d/2 ) = c g k+o(k) for a suitable constant c g > 0. Condition (3) implies that ϕ((1 − ∆ g ) d/2 ) ∈ L ϕ . Example 2.9. If (M, g) is a Riemannian manifold of dimension d, and ρ g denotes the geodesic distance, we can define an operator T : C ∞ c (M) → C ∞ (M) by (7) T f (x) := M f (x) − f (y) ρ g (x, y) d dV g ,

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Section: Acknowledgementsmentioning

confidence: 99%

“…We equipp the operator H with self-adjoint Robin type boundary conditions at x = a. See more in [20,Introduction]. It follows from [20, Section 1.1.3] that lim t→∞ N H (t) t 1/2 log t = 1 2π .…”

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confidence: 99%

Dixmier traces and residues on weak operator ideals

Goffeng

Usachev

2020

Journal of Mathematical Analysis and Applications

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“…Nesta subsec ¸ão apresentaremos o artigo [HLV17]. Nele investigamos o zeta-determinante de operadores de Sturm-Liouville com potenciais que possuem crescimento quadrático no infinito, ou seja, operadores da forma Pretendemos com este artigo iniciar uma discussão sobre estes operadores, paralela aos desenvolvimentos no contexto de operadores de Sturm-Liouville regularsingular, que por sua vez são motivados pela geometria de espac ¸os com singularidades do tipo c ônico.…”

Section: Zeta Determinantes De Operadores De Sturm-liouville Com Potenciais Quadráticos No Infinitounclassified

“…Na maioria dos artigos apresentados neste texto sistematizado as func ¸ões de Bessel possuem um papel central na obtenc ¸ão dos resultados. Na literatura são conhecidas diversas propriedades destas func ¸ões, em particular, em [HLV17] dedicamos uma sec ¸ão para a apresentac ¸ão das expans ões assint óticas conhecidas das func ¸ões de Bessel Modificadas.…”

Section: Zeta Determinantes De Operadores De Sturm-liouville Com Potenciais Quadráticos No Infinitounclassified

Análise Global e Teoria Espectral de pseudovariedades

Hartmann¹

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Apresento aqui alguns dos trabalhos desenvolvidos por mim, com ou sem colaborac ¸ão. Optei por não incluir todos, descrevendo apenas os que possuem uma temática comum, ou seja, os que estão relacionados ao estudo de invariantes espectrais sobre pseudovariedades.Gostaria, primeiramente, de explicar porque não comentarei sobre os trabalhos [BFH16, DOHV19, DAHA19]. O trabalho [BFH16] utiliza ferramentas de Topologia Algébrica, [DOHV19] está relacionado a Análise Funcional e [DAHA19] está ligado a Teoria de Singularidade. Todos não possuem conexão direta com os assuntos tratados nesse texto. Quanto aos trabalhos [DMHS09, DMHS12, HASP10, HAR14, HASP17A] também não os comentarei pois são desdobramentos diretos da minha tese [HAR09]. Os trabalhos não serão apresentados em ordem cronol ógica, e sim em uma ordem que acredito ser natural de modo a fazer a leitura fluir. Dessa forma no Capítulo 1 são discutidos os trabalhos relacionados ao zeta-determinante regularizado de um operador auto-adjunto, nele comentarei os trabalhos [HAR19, HLV17, HASP19]. No Capítulo 2 são discutidos os trabalhos relacionados com a extensão do Teorema de Cheeger-M üller para espac ¸os com singulares c ônicas, nele comentarei os artigos [HASP11, HASP12, HASP16, HASP17B]. No último capítulo, o Capítulo 3, apresento meus trabalhos relacionados a expansão assint ótica do trac ¸o do resolvente em pseudovariedade suavemente estratificadas, nele comento os trabalhos [HLV18A,

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Cheeger–Müller theorem on manifolds with cusps

Vertman

2018

Math. Z.

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We prove equality between the renormalized Ray-Singer analytic torsion and the intersection R-torsion on a Witt-manifold with cusps, up to an error term determined explicitly by the Betti numbers of the cross section of the cusp and the intersection R-torsion of a model cone. In the first step of the proof we compute explicitly the renormalized Ray-Singer analytic torsion of a model cusp in general dimension and without the Witt-condition. In the second step we establish a gluing formula for renormalized Ray-Singer analytic torsion on a general class of non-compact manifolds in any dimension that includes Witt-manifolds with cusps, but also scattering manifolds with asymptotically conical ends. In the final step, a Cheeger-Müller theorem on cusps follows by a combination of the previous explicit computation and the gluing formula.

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Zeta-determinants of Sturm–Liouville operators with quadratic potentials at infinity

Cited by 5 publications

References 23 publications

Dixmier traces and residues on weak operator ideals

Dixmier traces and residues on weak operator ideals

Análise Global e Teoria Espectral de pseudovariedades

Cheeger–Müller theorem on manifolds with cusps

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