Analytic Continuation of Resolvents of Elliptic Operators in a Multidimensional Cylinder
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Journal of Mathematical Sciences, Vol. 250, No. 2, October, 2020
ANALYTIC CONTINUATION OF RESOLVENTS OF ELLIPTIC OPERATORS IN A MULTIDIMENSIONAL CYLINDER D. I. Borisov
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Institute of Mathematics, UFRC RAS 112, Chernyshevskii St., Ufa 450008, Russia Bashkir State Pedagogical University 3a, October Revolution St., Ufa 450000, Russia University of Hradec Kr´ alov´e 62, Rokitansk´eho, Hradec Kr´ alov´e 50003, Czech Republic [email protected]
A. M. Golovina Bauman Moscow State Tchnical University 5/1. 2-ya Baumanskaya St., Moscow 105005, Russia [email protected]
A. I. Mukhametrakhimova Bashkir State Pedagogical University 3a, October Revolution St., Ufa 450000, Russia [email protected]
UDC 517.956+517.958
We consider the scalar second order elliptic differential operator in a multidimensional cylinder with the Dirichlet boundary condition. The coefficients of the operator are periodic on both outlets of the cylinder and are arbitrary in its finite part. We study a local analytic continuation of the bordered resolvent of the operator from the upper half-plane to the lower one with respect to the spectral parameter in a neighborhood of an interior point of the essential spectrum. It is shown that the size of the neighborhood depends only on geometric properties of the cylinder and the behavior of periodic components of coefficients of the operator. We introduce the notion of a resonance and formulate the corresponding boundary value problems. We describe the behavior of the bordered resolvent with respect to the spectral parameter in the neighborhood of the point of the essential spectrum. Bibliography: 14 titles. In mathematics, the notion of a resonance is connected with an analytic continuation of the resolvent of a given operator through the real axis with respect to the spectral parameter. The resonances are poles of such analytic continuations. We refer to [1, 2] and the references ∗
To whom the correspondence should be addressed.
Translated from Problemy Matematicheskogo Analiza 105, 2020, pp. 67-87. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2502-0260
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therein for an information about analytic continuations of different operators and properties of resonances. Resonances of operators with distant perturbations were studied in [3]–[7]. While studying particular models, it turned out that there are resonances with unexpected properties. General models with distant perturbations have not been sufficiently studied yet, and the study of such models is actual, but still remains an open question. The most suitable is the model of a general elliptic operator with several distant perturbations in an infinite multidimensional cylinder, and the first step of the research consists in introducing an appropriate definition of a resonance, based on a suitable analytic continuation of the resolvent. It is necessary not only to establish the existence of such an analytic continuation in some neighborhood of a given point of the essential spectrum, but also to control the size of the neighborhood depending
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