An Invitation to Random Schr¨odinger operators - FernUniversität in ...

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〈ϕ, µ(M)ψ〉 = µ ϕ,ψ (M) (3.40)
The functional calculus can be implemented for unbounded self adjoint operators
as well. For such operators the spectrum is always a closed set. It is compact only
for bounded operators.
We will use the functional calculus virtually everywhere throughout this paper. For
example, it gives meaning to the operator e −itH used in (2.4). We will look at the
projection valued measures χ M (A) more closely in chapter 7.
3.3. Some more functional analysis.
In this section we recall a few results from functional analysis and spectral theory
and establish notations at the same time. In particular, we discuss the min-max
principle and the Stone-Weierstraß theorem.
Let A be a selfadjoint (not necessarily bounded) operator on the (separable) Hilbert
space H with domain D(A). We denote the set of eigenvalues of A by ε(A). Obviously,
any eigenvalue of A belongs to the spectrum σ(A). The multiplicity of an
eigenvalue λ of A is the dimension of the eigenspace {ϕ ∈ D(A); Aϕ = λϕ}
associated to λ. If µ is the projection valued spectral measure of A, then the
multiplicity of λ equals tr µ({λ}). An eigenvalue is called simple or non degenerate
if its multiplicity is one, it is called finitely degenerate if its eigenspace
is finite dimensional. An eigenvalue λ is called isolated if there is an ε > 0
such that σ(A) ∩ (λ − ε, λ + ε) = {λ}. Any isolated point in the spectrum
is always an eigenvalue. The discrete spectrum σ dis (A) is the set of all isolated
eigenvalues of finite multiplicity. The essential spectrum σ ess (A) is defined by
σ ess (A) = σ(A) \ σ dis (A).
The operator A is called positive if 〈φ, Aφ〉 ≥ 0 for all φ in the domain D(A), A
is called bounded below if 〈φ, Aφ〉 ≥ −M〈φ, φ〉 for some M and all φ ∈ D(A).
We define
µ 0 (A) = inf {〈φ, Aφ〉 ; φ ∈ D(A), ‖φ‖ = 1} (3.41)
and for k ≥ 1
µ k (A) = sup
ψ 1 ,...,ψ k ∈H
inf {〈φ, Aφ〉 ; φ ∈ D(A), ‖φ‖ = 1, φ ⊥ ψ 1 , . . . , ψ k }
(3.42)
The operator A is bounded below iff µ 0 (A) > −∞ and µ 0 (A) is the infimum of
the spectrum of A.
If A is bounded below and has purely discrete spectrum (i.e. σ ess (A) = ∅), we can
order the eigenvalues of A in increasing order and repeat them according to their
multiplicity, namely
E 0 (A) ≤ E 1 (A) ≤ E 2 (A) ≤ . . . . (3.43)
If an eigenvalue E of A has multiplicity m it occurs in (3.43) exactly m times.
The min-max principle relates the E k (A) with the µ k (A).