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**Example text**

If ma (λ) = 1, then the eigenvalue is called algebraically simple or simple. It is true that ma (λ) ≥ mg (λ). An eigenvalue is called semi-simple if mg (λ) = ma (λ). e. has a countably infinite dense subset, then the following hold: (a) 0 ∈ σ(L), (b) if λ ∈ σ(L) with λ 0, then λ is isolated and ma (λ) < ∞, (c) σ(L) is a countable set, and the only possible accumulation point is λ = 0. If λ is an isolated eigenvalue, let C ⊂ C be a simple closed positively oriented curve surrounding λ that does not intersect the spectrum of L and whose interior contains no other points in σ(L).

7) p(0) = p(π), ∂x p(0) = ∂x p(π), the operator L is still self-adjoint in the weighed inner product over the space 2 Hper [0, π] = {u ∈ H 2 [0, π] : u(0) = u(π), ∂x u(0) = ∂x u(π)}. 1 must be revised; in particular, the eigenvalues need not be simple, and the correspondence between the eigenfunction and its number of zeros diﬀers [200]. 2. 4) with periodic bound2 [0, π]. 7) in the space Hper valued and can be enumerated in a descending order λ0 > λ1 ≥ λ2 ≥ λ3 ≥ λ4 > · · · , lim λn = −∞. n→+∞ The eigenfunction pj (x) associated with the eigenvalue λj for j = 0, 1, 2, .

4 Additional Reading Further details regarding the spectrum of Sturmian operators can be found in Carmona and Lacroix [45], Weidmann [277, 278]. Chapter 3 Essential and Absolute Spectra The goal of this chapter is the characterization of the essential spectrum and Fredholm indices of two classes of linear diﬀerential operators on unbounded domains. The first class is comprised of nth-order diﬀerential operators with spatially varying coeﬃcients that tend at an exponential rate to constant values at ±∞.