Self-Adjoint Extensions with Friedrichs Lower Bound
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Complex Analysis and Operator Theory
Self-Adjoint Extensions with Friedrichs Lower Bound Matteo Gallone1
· Alessandro Michelangeli2
Received: 28 April 2020 / Accepted: 29 August 2020 © The Author(s) 2020
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. Keywords Lower semi-bounded symmetric operators · Self-adjoint extensions · Friedrichs extensions Mathematics Subject Classification 47B15 · 47B25 · 47N50 · 81Q10
1 Motivation We start with a familiar example. In the Hilbert space H = L 2 (0, 1) let us consider the densely defined, closed, and symmetric operator
Communicated by Daniel Aron Alpay. Partially supported by the Istituto Nazionale di Alta Matematica (INdAM) and the Alexander von Humboldt foundation. This article is part of the topical collection “Spectral Theory and Operators in Mathematical Physics” edited by Jussi Behrndt, Fabrizio Colombo and Sergey Naboko.
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Matteo Gallone [email protected] Alessandro Michelangeli [email protected]
1
Dipartimento di Matematica, Università degli Studi di Milano, via Cesare Saldini 50, 20133 Milan, Italy
2
Institute for Applied Mathematics, and Hausdorff Center for Mathematics, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany 0123456789().: V,-vol
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S = −
M. Gallone, A. Michelangeli
d2 , D(S) = H02 (0, 1) = dx 2
f (0) = 0 = f (1) f ∈ H 2 (0, 1) . (1.1) f (0) = 0 = f (1)
S is actually 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) :=
f, Rf ∈ [−∞, +∞) f ∈D (R) f 22 inf
(1.2)
f =0
the largest lower bound of R. When m(R) > −∞ one says that R is semi-bounded from below. Poincaré inequality implies that S is semi-bounded from below with m(S) = π 2 .
(1.3)
Now, S is symmetric but not self-adjoint, for S∗ = −
d2 , dx 2
D(S ∗ ) = H 2 (0, 1) .
(1.4)
Thus, S admits a multiplicity (in fact, a four-real-parameter family) of distinct selfadjoint extensions, which are all restrictions of S ∗ . Among them, the Friedrichs extension S F is the one with domain D(S F ) = H 2 (0, 1) ∩ H01 (0, 1) =
f ∈ H 2 (0, 1) f (0) = 0 = f (1) , (1.5)
namely the Dirichlet (negative) Laplacian. Let us recall that abstractly speaking the Friedrichs extension of a lower semi-bounded symmetric operator S is the only selfadjoint extension of S with the property D(S F ) ⊂ D[S] ,
(1.6)
that is, with operator domain contained in the form domain of S. Here and in the following D[R] denotes the form domain of a lower semi-bounded symmetric operator R, or also of a self-adjoint operator R (see, e.g., [11, Chapt. 10]); in the present case D[S] = D(S)
H 1
= H01 (0, 1) ,
(1.7)
and obviously D[S] = D[S F
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