Abstract
In this chapter we recall, with more details, properties of unbounded symmetric and self-adjoint operators and their connection with quadratic forms. We give a brief review of some standard facts on the extension theory of symmetric operators and the general theory of quadratic forms. In particular, we formulate and prove the theorem that gives the basic criterion for self-adjointness. These facts will be used in the sequel. We recall also the well-known theorems on operator representations of quadratic forms. A more extensive treatment of the theory can be found in the well-known books by N.I. Akhiezer and I.M. Glazman [32], by T. Kato [107], and by M. Reed and B. Simon [169, 170].
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