Spectrum of the complex Laplacian on product domains

Chakrabarti, Debraj

doi:10.1090/s0002-9939-10-10522-x

Cited by 8 publications

(12 citation statements)

References 18 publications

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“…For the details on the general definition of ∂ E and its properties, we refer to [24,16,14,8]. We note that equation (5.2) has also appeared in [4]. In the following Theorem 5.1, we will consider the case where all ∆ϕ j define nontrivial doubling measures.…”

Section: Decoupled Weightsmentioning

confidence: 99%

On some spectral properties of the weighted ∂¯-Neumann operator

Berger¹,

Haslinger²

2019

Kyoto J. Math.

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We study necessary conditions for compactness of the weighted ∂-Neumann operator on the space L 2 (C n , e −ϕ ) for a plurisubharmonic function ϕ. Under the assumption that the corresponding weighted Bergman space of entire functions has infinite dimension, a weaker result is obtained by simpler methods. Moreover, we investigate (non-) compactness of the ∂-Neumann operator for decoupled weights, which are of the form ϕ(z) = ϕ1(z1) + · · · + ϕn(zn). More can be said if every ∆ϕj defines a nontrivial doubling measure.2010 Mathematics Subject Classification. Primary 32W05; Secondary 30H20, 35N15, 35P10.

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Section: Decoupled Weightsmentioning

confidence: 99%

On some spectral properties of the weighted ∂¯-Neumann operator

Berger¹,

Haslinger²

2019

Kyoto J. Math.

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“…However there are a few cases where Condition R can be established on a domain with generic corners by elementary means. In [14,13], the following was proved: if D 1 ⊂ C n1 , D 2 ⊂ C n2 are bounded pseudoconvex domains (no assumption of generic corners on the boundary) such that each of them satisfies Condition R, then so does their product. Here on the other hand there is no assumption of pseudoconvexity.…”

Section: Some Examplesmentioning

confidence: 99%

Condition R and holomorphic mappings of domains with generic corners

Chakrabarti¹,

Verma²

2013

Illinois J. Math.

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A piecewise smooth domain is said to have generic corners if the corners are generic CR manifolds. It is shown that a biholomorphic mapping from a piecewise smooth pseudoconvex domain with generic corners in complex Euclidean space that satisfies Condition R to another domain extends as a smooth diffeomorphism of the respective closures if and only if the target domain is also piecewise smooth with generic corners and satisfies Condition R. Further it is shown that a proper map from a domain with generic corners satisfying Condition R to a product domain of the same dimension extends continuously to the closure of the source domain in such a way that the extension is smooth on the smooth part of the boundary. In particular, the existence of such a proper mapping forces the smooth part of the boundary of the source to be Levi degenerate.

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“…The Cauchy-Riemann equations on product domains have been studied previously in [Cha10; CS11; Ehs07; Fu07; Kra88]. In [Cha10], Chakrabarti computes the spectrum of for X × Y , the product of two Hermitian manifolds. If we denote, for the moment, the complex Laplacian on the (p, q) forms on X × Y by X×Y p,q , then its spectrum according to [Cha10] is σ( X×Y p,q ) = p +p =p q +q =q σ( X p ,q ) + σ( Y p ,q ) .…”

Section: Introductionmentioning

confidence: 99%

Essential spectra of tensor product Hilbert complexes and the ∂‾-Neumann problem on product manifolds

Berger

2016

Journal of Functional Analysis

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Abstract. We investigate tensor products of Hilbert complexes, in particular the (essential) spectrum of their Laplacians. It is shown that the essential spectrum of the Laplacian associated to the tensor product complex is computable in terms of the spectra of the factors. Applications are given for the ∂-Neumann problem on the product of two or more Hermitian manifolds, especially regarding (non-) compactness of the associated ∂-Neumann operator.

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Spectrum of the complex Laplacian on product domains

Cited by 8 publications

References 18 publications

On some spectral properties of the weighted ∂¯-Neumann operator

On some spectral properties of the weighted ∂¯-Neumann operator

Condition R and holomorphic mappings of domains with generic corners

Essential spectra of tensor product Hilbert complexes and the ∂‾-Neumann problem on product manifolds

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