By William G Faris
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Extra resources for Self-adjoint operators
Example text
4 be a Hermitian operator. Then A is A is also Hermitian. 5 . and Let is closed, A Ai' A be Hermitian and is closable and A A c A'~ . be self-adjoint with 1 Then Proof: A = A~ , A 1 1 Since is closed. Thus A c A1 Since AcA , 1 A~ c A'" , that is, A c A'~ 1 A Hermitian operator if A is said to be essentially self-adjoint A is self-adjoint. 6 . If A A{' A is essentially self- is Hermitian (A'" c A;";) • is essentially self-adjoint, then it has a unique self-adjoint extension. Proof: If Al is self-adjoint' then A cAlc A'~ = A* = A - 33 - EXAMPLES (l) Let domain H = L2 (Rn,dx) and let A be the Laplace operator with functions with compact V(A) support.
H implies Af ~ is said to be 0 Since A is linear, if it is positivity preserving it is also order preserving, that is, f ~ g implies ~ Af Ag . L2 Notice that the set of positive functions in cone. is a closed Thus the property of being positivity preserving is preserved under strong limits. This fact will be used in the proof of the following proposition. 1 Let operator acting in H preserving for all c on M . Set V Assume that =U preserving for all Proof: 0 ~ uk Let ~ k uk and H c and -r easy to verify that Let is positivity 0 w satisfies condition A H + V .
A preliminary step in this construction is the construction of a theory on a finite interval, essentially the quantum theory of a non-linear vibrating string. We begin with the construction of the field. periodic interval of length multiples of (2n/L) and let Let L r+ r Consider a be the set of all integer be the set of strictly positive - 58 - 1 integer multiples of (2n/L) . The functions (l/L) 2 and 1 J. (2/L) 2 cos(kx) (2/L) 2 sin(kx) , where and k r ranges over + form an orthonormal basis for the functions on the interval.