Full asymptotic expansion of the heat trace for non–self–adjoint elliptic cone operators

Gil, Juan B.

doi:10.1002/mana.200310020

Cited by 23 publications

(1 citation statement)

References 42 publications

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“…Zeta functions have also been studied for more general 'cone operators', which generalize regular singular operators (see, e.g., Gil [29]). For recent and ongoing work involving resolvents of general self-adjoint extensions of cone operators, which is the first step to a full understanding of zeta functions, see Gil et al [30,31] and Coriasco et al [17].…”

Section: Self-adjoint Extensionsmentioning

confidence: 99%

The ubiquitous ζ-function and some of its ‘usual’ and ‘unusual’ meromorphic properties

Kirsten

Loya

Park

2008

J. Phys. A: Math. Theor.

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Abstract. In this contribution we announce a complete classification and new exotic phenomena of the meromorphic structure of ζ-functions associated to conic manifolds proved in [37]. In particular, we show that the meromorphic extensions of these ζ-functions have, in general, countably many logarithmic branch cuts on the nonpositive real axis and unusual locations of poles with arbitrarily large multiplicity. Moreover, we give a precise algebraic-combinatorial formula to compute the coefficients of the leading order terms of the singularities.

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Section: Self-adjoint Extensionsmentioning

confidence: 99%

The ubiquitous ζ-function and some of its ‘usual’ and ‘unusual’ meromorphic properties

Kirsten

Loya

Park

2008

J. Phys. A: Math. Theor.

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Meromorphic symbolic structures for boundary value problems on manifolds with edges

Donno

Schulze

2006

Mathematische Nachrichten

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Key words Boundary problems on corner manifolds, symbolic structures with asymptotics, kernel cut-off and holomorphic symbols MSC (2000) 35J70, 35S05, 58J40We investigate the ideal of Green and Mellin operators with asymptotics for a manifold with edge-corner singularities and boundary which belongs to the structure of parametrices of elliptic boundary value problems on a configuration with corners whose base manifolds have edges. IntroductionParametrices of elliptic boundary value problems for differential operators on a manifold with smooth boundary belong to a pseudo-differential calculus with a symbolic hierarchy (interior and boundary symbols) and with typical contributions from the boundary (Green, trace, and potential operators), cf. Boutet de Monvel [1]. Similar structures may be obtained for the case of manifolds with geometric singularities, e.g., conical points, edges, corners, etc., as they are natural in a number of applications. Such problems belong to the analysis of operators on stratified and non-compact spaces. However, as is known from the analysis for conical and edge-singularities, answers are far from being straightforward. This concerns, in particular, the regularity and asymptotics of solutions near the singularities, the nature of extra conditions along lower-dimensional skeleta (including topological obstructions for their existence), and the description of analogues of Green functions (Green operators). Another experience from the known scenario for conical singularities is that meromorphic operator functions operating on the base of corners play a crucial role, both as the conormal symbolic structure and for evaluating the asymptotics of solutions and relative indices under changing of weights. In the case of corner singularities, where the base itself has conical points or edges, this has to be combined with an edge symbolic calculus. These ingredients contribute to Green (plus Mellin) operators. The program of this paper is to characterize the corresponding algebra of Green (plus Mellin) operators for the case of boundary value problems on a manifold with edges and corners.The geometry near the corner points is that of a local cone where the base is a manifold with edges and boundary (that means, our configuration has edges with conical singularities and in addition a boundary). Similar structures for the case without boundary have been investigated in Schulze [26]. Other simpler special cases are manifolds with smooth edges and boundary; this is studied systematically in the monograph of Kapanadze and Schulze [9], motivated by applications in crack theory. In particular, in such models it is interesting to analyse the mechanism of how solutions to elliptic boundary value problems "acquire" asymptotics close to the singularities. By Kondratyev's work [10] this is a famous story for conical singularities with smooth base manifolds. Later on many other special cases have been studied, cf. the references in [26] or [9].For corner singularities the asymptotic information is complicated...

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On the completeness of the generalized eigenfunctions of elliptic operators on manifolds with conical singularities

Krainer

2010

Mathematische Nachrichten

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Full asymptotic expansion of the heat trace for non–self–adjoint elliptic cone operators

Cited by 23 publications

References 42 publications

The ubiquitous ζ-function and some of its ‘usual’ and ‘unusual’ meromorphic properties

The ubiquitous ζ-function and some of its ‘usual’ and ‘unusual’ meromorphic properties

Meromorphic symbolic structures for boundary value problems on manifolds with edges

On the completeness of the generalized eigenfunctions of elliptic operators on manifolds with conical singularities

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