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Let us introduce a new spectral parameter k such that λ(k) = − ω1 k 2 + 14 . From now on we consider the case where m is a Schwartz class function, and m(x, 0)+ω > 0. Then m(x, t) + ω > 0 for all t . The spectral picture of (10) is : 38 A. I. Ivanov continuous spectrum: k — real; discrete spectrum: ﬁnitely many points kn = ±iκn , n = 1, . . , N where κn is real and 0 < κn < 1/2. Eigenfunctions: for all real k = 0 a basis in the space of solutions can be introduced, ﬁxed by ¯ k), such that its asymptotic when x → ∞: ψ(x, k) and ψ(x, ψ(x, k) = e−ikx + o(1), x → ∞.

Tu × ts ) dsdu on a unit sphere is just the solid angle δΩ. Substituting Eqs. (26) and (27) in Eq. (25), we get δΦinc = δΦF W + δΩ. (28) This is the general geometric relationship between the two types of geometric phases, and it is valid for all smooth curve evolutions. Returning to Eqs. (19) and (20) with ∆t = 0, we get Ku = gs − hτ, τ 0 = (hs + gτ )/K. (29) So far, our analysis has been quite general. We must now ﬁnd g, h and τ 0 in such a way that Eqs. (17) and (18) are obtained. In turn, this would imply the equivalence of the Schr¨ odinger equation (15) and the curve evolution equations (1) and (5), as already explained.

9) The total phase is then given by the following expression: Φ=− u2 s0 du u1 −s0 Rs ds. (10) Using the s and u derivatives of t, we get that ts ∧ tu = kht. This allows us to give a geometrical interpretation of the total phase: Φ=− u2 s0 du u1 −s0 t · (tu ∧ ts )ds. (11) The expression t · (tu ∧ ts )dsdu represents the element of area on the unit sphere. Fermi-Walker parallel transport 45 A. Quantum mechanical phase: Berry’s phase Note that the quantity R can be expressed using the complex vector N: R = − 2i N ∗ ·Nu With this the expression for the geometric phase becomes: Φ= i 2 s0 −s0 ds ∂ ∂s u2 N ∗ ·Nu du.