Selfadjointness of elliptic differential operators in 𝐿₂(𝐺), and correction potentials

Brusentsev, A.

doi:10.1090/S0077-1554-04-00144-X

Selfadjointness of elliptic differential operators in $L_2(G)$, and correction potentials
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by A. G. Brusentsev
Translated by: Michael Grinsfeld

Trans. Moscow Math. Soc. 2004, 31-61

DOI: https://doi.org/10.1090/S0077-1554-04-00144-X

Published electronically: October 1, 2004

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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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