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arXiv · 1812.11399

Heat kernel for higher-order differential operators in Euclidean space

W. Wachowski·P. I. Pronin
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Abstract

We consider heat kernel for higher-order operators with constant coefficients in $d$-dimensio\-nal Euclidean space and its asymptotic behavior. For arbitrary operators which are invariant with respect to $O(d)$-rotations we obtain exact analytical expressions for the heat kernel and Green functions in the form of infinite series in Fox--Wright psi functions and Fox $H$-functions. We investigate integro-differential relations and the asymptotic behavior of the functions $ \mathcal{E}_{ν, α}(z)$, in terms of which the heat kernel of $O(d)$-invariant operators are expressed. It is shown that the obtained expressions are well defined for non-integer values of space dimension $d$, as well as for operators of non-integer order. Possible applications of the obtained results in quantum field theory and the connection with fractional calculus are discussed.

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W. Wachowski, P. I. Pronin. 2018-12-29. Heat kernel for higher-order differential operators in Euclidean space. https://arxiv.org/abs/1812.11399

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