By Pianigiani G.

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**Example text**

We will come back on this question later on, when referring to the case of Schr¨odinger equation with quasi-periodic potential. In the above discussion of the m-function, the dependence of m on the endpoint a and the boundary condition α was not considered. From now on we will restrict the notation mα (λ) to the case a = 0 and the notation m(λ) to the case α = π/2. In this case we have that the boundary conditions become ϕ(a; λ) = 1, ϕ (a; λ) = 0 ψ(a; λ) = 0, ψ (a; λ) = 1, and therefore ϕ(·, λ) and ψ(·, λ) are the normalized solutions at a.

Subresolution of the identity) of H on (Ω, B) if: (i) E(Ω) = I ( resp. (i’) E(∅) = 0). (ii) E(∪n≥1 An ) = disjoint. n≥1 E(An ), whenever {An ; n ≥ 1} is a sequence in B whose elements are The convergence in item (ii) has to be understood in the sense of the strong convergence of operators. This means that for each fixed f ∈ H, the series n≥1 E(An )f converges in H to E(∪n≥1 An )f . Hence for each fixed f ∈ H, the function A ∈ B → E(A)f ∈ H is countably additive, and this is usually called a countably additive measure.

10) are square integrable near +∞ (resp. −∞). Otherwise it is said to be in the limit point case. In order to prove the essential self-adjointness we will do the following assumption, which is typical in the context of Schr¨odinger operators with bounded potential. We will assume that V is a measurable real function such that there exist constants a > 0 and C > 0 satisfying V (t) ≥ −a(t2 + 1), |t| > C. 11) This condition is satisfied when V (t) is bounded, which is the case if V (t) = Q(ωt + φ), being Q continuous on Td .