By Gladwell G., Morassi A. (eds.)

ISBN-10: 3709106958

ISBN-13: 9783709106952

The papers during this quantity current an summary of the final points and sensible purposes of dynamic inverse tools, throughout the interplay of a number of themes, starting from classical and complex inverse difficulties in vibration, isospectral structures, dynamic tools for structural identity, energetic vibration regulate and harm detection, imaging shear stiffness in organic tissues, wave propagation, to computational and experimental points suitable for engineering difficulties.

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Therefore, we try to ﬁnd zi in the form zi = wi + ri . (143) Replacing this expression in (135)–(136), the perturbation ri solves the initial value problem Classical Inverse Eigenvalue Problems 49 ⎧ ⎪ ⎨ ri + (λi − q(x))ri = −ζi (wi + ri ), in (0, 1), ri (0) = 0, ⎪ ⎩ r (0) = 0. i (144) (145) (146) Then, by well-known results (see P¨oschel and Trubowitz (1987), Chapter 1, Theorem 2) we have ri (x) = −ζi x 0 (y1 (t)y2 (x) − y2 (t)y1 (x))(wi (t) + ri (t))dt, (147) where {y1 , y2 } is a set of fundamental solutions of the diﬀerential equation y + (λi − q)y = 0 in (0, 1) satisfying the initial conditions y1 (0) = y2 (0) = 1 and y1 (0) = y2 (0) = 0.

120) 46 A. Morassi The subtle point is now to realize that the right hand side of (120) is closely related to the Mittag-Leﬄer expansion of the function g(x) = v(x) x 0 f (y)w(y)dy + w(x) ω(λ) 1 x f (y)v(y)dy , (121) where v(x) is solution of (96)-(98) with q replaced by q, and w(x) is solution of (84)-(86). In fact, we can represent g(x) in Mittag-Leﬄer series distinguishing between residues related to poles belonging to Λ0 and to poles belonging to Λ: un (x) g(x) = Λ0 x 0 f (y)wn (y)dy + zn (x) ω (λn )(λ − λn ) vn (x) + x 0 Λ 1 x f (y)vn (y)dy f (y)wn (y)dy + wn (x) ω (λn )(λ − λn ) 1 x + f (y)vn (y)dy , (122) where, to simplify the notation, we deﬁne un = v(x, λn ), zn = w(x, λn ), n ∈ Λ0 .

20) y(x, λn ) = √ +O n2 λn Recalling that gn (x) = estimate y(x,λn y(x,λn ) L2 gn (x) = √ we obtain the asymptotic eigenfunction 2 sin(nπx) + O 1 n , (21) 34 A. Morassi which holds uniformly on bounded subsets of [0, 1] × L2 (0, 1) as n → ∞. Finally, by iterating the above procedure, the eigenvalue estimate (18) can be improved to obtain λn = (nπ)2 + 3 1 0 q(x)dx − 1 cos(2nπx)q(x)dx + O 0 1 n , as n → ∞. 1 Symmetric Potential and Dirichlet Boundary Conditions Let us consider the Dirichlet eigenvalue problem y (x) + λy(x) = q(x)y(x), y(0) = 0 = y(1), in (0, 1), (23) (24) where q ∈ L2 (0, 1) is a real-valued potential.

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