Local Commutator Estimates
In this chapter, we will examine a number of theorems about operators H which follow from the Mourre estimate, an estimate which says that a commutator (H, iA) is positive in some sense. The ideas in this chapter can be traced back to Putnam(289), Kato(19
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In this chapter, we will examine a number of theorems about operators H which follow from the Mourre estimate, an estimate which says that a commutator [H, iA] is positive in some sense. The ideas in this chapter can be traced back to Putnam [289], Kato [191] and Lavine [225] for theorems on the absence of singular spectrum, and to Weidmann [367] and Kalf [189] for theorems on absence of positive eigenvalues. All this earlier work applied to rather restricted classes of potentials. It was Mourre [256], in a brilliant paper, who realized that by only requiring localized estimates, one could deal with fairly general potentials. He developed an abstract theory which he was able to apply to 2- and 3-body Schrödinger operators. Perry, Sigal and Simon [281] showed that his ideas could handle N-body Schrödinger operators. In Sect. 4.1, we prove Putnam’s theorem on the absence of singular spectrum, and introduce the Mourre estimate. We then give some examples of Schrödinger operators for which a Mourre estimate holds, deferring the proof of the estimate for N-body Schrödinger operators until Sect. 4.5. In Sect. 4.2, we prove the virial theorem and show how this, together with a Mourre estimate, can give information about the accumulation of eigenvalues. In Sect. 4.3, we prove a variant of the theorem of Mourre [256] on absence of singular spectrum. In Sect. 4.4, we present theorems of Froese and Herbst [114], and Froese, Herbst, HoffmannOstenhof and Hoffmann-Ostenhof [116] on L 2 -exponential bounds for eigenfunctions of Schrödinger operators which imply that N-body Schrödinger operators have no positive eigenvalues.
4.1 Putnam’s Theorem and the Mourre Estimate Commutator methods appear in a simple form in Putnam’s theorem, where positivity of a commutator is used to prove absolute continuity of spectrum. We first give a convenient criterion for the absolute continuity of spectrum. Proposition 4.1. Suppose H is a self-adjoint operator, and R(z) = (H − z)−1 . Suppose for each ϕ in some dense set there exists a constant, C(ϕ) < ∞ such that lim sup ϕ, Im R(μ + iε)ϕ ≤ C(ϕ) . ε↓0 μ∈(a,b)
Then H has purely absolutely continuous spectrum in (a, b).
4.1 Putnam’s Theorem and the Mourre Estimate
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4. Local Commutator Estimates
The Mourre estimate can be thought of as a weak form of hypothesis (4.1). In the Mourre estimate, H and A can be unbounded, which is crucial for applications to Schrödinger operators. Moreover, the Mourre estimate is local in the spectrum of H. Thus, we will be able to prove absolute continuity of the spectrum of H away from eigenvalues without proving (as Putnam’s theorem does) that eigenvalues do not exist. Before describing the Mourre estimate, we need some definitions. We first define a scale of spaces associated with a self-adjoint operator H. Definition 4.3. Given a self-adjoint operator H acting in a Hilbert space H, define H+2 := D(H) with the graph norm ψ +2 = (H + i)ψ . Similarly, define H+1 := D(|H|1/2 ) with its graph norm. Define H−2 , and H−1 to be the dual spaces of H+2 and H+1 ,
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