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Transactions of the Moscow Mathematical Society

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Selfadjointness of elliptic differential operators in $L_2(G)$, and correction potentials


Author: A. G. Brusentsev
Translated by: Michael Grinsfeld
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 65 (2004).
Journal: Trans. Moscow Math. Soc. 2004, 31-61
MSC (2000): Primary 35J15; Secondary 35J10, 58J05
DOI: https://doi.org/10.1090/S0077-1554-04-00144-X
Published electronically: October 1, 2004
MathSciNet review: 2193436
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Abstract: We consider the question of the essential selfadjointness of a symmetric second order elliptic operator $L$ of general form in the space $L_2(G)$ $\left(D_L=C_0^\infty(G)\right)$, where $G$  is an arbitrary open set in $R^n$. The main idea is that using the matrix $A(x)$ of the highest order coefficients of $L$ and the domain $G$, it is possible to construct a function $q_A(x)$ such that the essential selfadjointness of $\Bar{L}$ follows from the semiboundedness of the operators $L$ and $L-q_A(x)$. The function $q_A(x)$ is called the correction potential, and we suggest a number of procedures for its construction. We develop a technique which, given a correction potential allows us to establish criteria for the selfadjointness of an elliptic operator in terms of the behaviour of its coefficients. These general results are applied to the Schrödinger operator, which for $G\ne R^n$ leads to new assertions that generalise a number of known theorems.


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Additional Information

A. G. Brusentsev
Affiliation: V. G. Shukhov Belgorod State Technological University, Belgorod, Russia
Email: brisentsev@mail.ru

DOI: https://doi.org/10.1090/S0077-1554-04-00144-X
Published electronically: October 1, 2004
Article copyright: © Copyright 2004 American Mathematical Society

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