CHAPTER 1 Preliminaries We start by reducing the main equation Lu = Au + Bu into a simpler form. Then, we define a family of operators L , their adjoint L∗, and prove a Green’s formula. The operators L will be extensively used in the next chapter. Let λ = a + ib R+ + iR∗ and define the vector field L by (1.1) L = λ ∂t ir ∂r . For A Ck(S1, C), with k Z+, set A0 = 1 0 A(t)dt, ν = 1 Im A0 λ + Im A0 λ where for x R, [x] denotes the greatest integer less or equal than x. Hence, ν [0, 1). Define the function m(t) = exp it + i Im A0 λ t + 1 λ t 0 (A(s) A0)ds . Note that m(t) is 2π-periodic. The following lemma is easily verified. Lemma 1.1. Let A, B Ck(S1, C) and m(t) be as above. If u(r, t) is a solution of the equation (1.2) Lu = A(t)u + B(t)u then the function w(r, t) = u(r, t) m(t) solves the equation (1.3) Lw = λ Re A0 λ w + C(t)w where C(t) = B(t) m(t) m(t) . In view of this lemma, from now on, we will assume that Re A0 λ = 0 and deal with the simplified equation (1.4) Lu = −iλνu + c(t)u where ν [0, 1) and c(t) Ck(S1, C). Consider the family of vector fields (1.5) L = λ ∂t ir ∂r 5
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