By Z. S. Marchenko, V. A. Agranovich

Half 1. The boundary-value challenge with no singularities --

I. specific recommendations of the method with out singularities --

II. The spectrum and scattering matrix for the boundary-value challenge with out singularities --

III. the elemental equation --

IV. Parseval's equality --

V. The inverse challenge --

Part 2. The boundary-value challenge with singularities --

VI. distinct transformation operators --

VII. Spectral research of the boundary-value challenge with singularities --

VIII. Reconstruction of the singular boundary-value challenge from its scattering information --

Appendix I. at the attribute homes of the scattering information of the boundary-value challenge with out singularities --

Appendix II. Refinement of yes inequalities.

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1) and the fact that f'a(x) dx s a1(a) < oo, a j :z;l+aa2(x) J a(x) dx J tl+a IV(t) I dt dx s Also, for b (a>O). t IV(t) I dt + -1 ·. b ei+a jV(t) I dt+ 2µ 2µ ::s - f jV(t) I dt+-1 fb xiH jV(x) I dx. t jV(t) I dt < 00° (a> 0). ) I dx< oo (a>O). ) (Im A. = 0, A. ¥=: 0). 2) and has the required asymptotic behavior at infinity. ) for sufficiently large J. A. A. ) dt. A. ) = I - 2 f f . A. ) dt. h It is easy to show by· the method of successive approximations that a solution to this equation exists in the interval h s x < oo.

2) with respect to x and. A. A. - (Im< 0). 14) The validity of these two expressions is proved in the same way, and we shall give the reasoning only for the latter one. A. 14) is uniformly convergent for x and A. 14) for x > 0 and Im A. < 0. 9), we obtain the following three estimates: I K(x, x) ·I for Im A. V(u)l_du+2ca(x) x ae;t)dt= [ ! = -µ < 0, l JtKx(x, t)e-iAI dt Is ! ,s; 2 s jV(s) I ds = 2a1( +) :s 2a~ (0). 14), it is apparent that to complete the proof of the lemma it suffices to show that lim xa(x) x-+o = 0.

At infinity stated in the theorem. I. A. A. A. A. x ·s K(t, s)e-iA(s-t) ds ] dt. A. A. J K(t, s)e-iA(s-t) ds dt. = -µ < 0. _ ev,(x)a2(x). 1) and the fact that f'a(x) dx s a1(a) < oo, a j :z;l+aa2(x) J a(x) dx J tl+a IV(t) I dt dx s Also, for b (a>O). t IV(t) I dt + -1 ·. b ei+a jV(t) I dt+ 2µ 2µ ::s - f jV(t) I dt+-1 fb xiH jV(x) I dx. t jV(t) I dt < 00° (a> 0). ) I dx< oo (a>O). ) (Im A. = 0, A. ¥=: 0). 2) and has the required asymptotic behavior at infinity. ) for sufficiently large J. A. A. ) dt.