Abstract
We produce a simple criterion and a constructive recipe to identify those self-adjoint extensions of a lower semi-bounded symmetric operator on Hilbert space which have the same lower bound as the Friedrichs extension. Applications of this abstract result to a few instructive examples are then discussed.
Highlights
And in the following D(R) denotes the domain of the operator R acting on H, and if R is symmetric we denote by m(R) := inf f ∈D(R) f =0 f, Rf f the largest lower bound of R
In this note we present a simple criterion and a constructive recipe to identify those self-adjoint extensions of a lower semi-bounded symmetric operator on Hilbert space which have the same lower bound as the Friedrichs extension
Let us denote by S(K) the collection of all self-adjoint operators defined in Hilbert subspaces of a given Hilbert space K: Theorem 2.1 states that the self-adjoint extensions of S are all of the form ST for some T ∈ S(ker S∗)
Summary
S is the operator closure of the negative Laplacian defined on C0∞(0, 1). Here and in the following D(R) denotes the domain of the operator R acting on H, and if R is symmetric we denote by m(R) := inf f ∈D(R) f =0 f , Rf f. This occurrence is well known: a lower semi-bounded symmetric operator may admit self-adjoint extensions other than the Friedrichs, with the same bottom of the Friedrichs spectrum. This is not typical of symmetric operators with deficiency index 2 only, as was the case for S here. In this note we present a simple criterion and a constructive recipe to identify those self-adjoint extensions of a lower semi-bounded symmetric operator on Hilbert space which have the same lower bound as the Friedrichs extension.
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