Editing Inverse problem (section)

==== Inverse scattering problems ====

Whereas linear inverse problems were completely solved from the theoretical point of view at the end of the nineteenth century {{Citation needed|reason=What specific date ? Is there a publication that marks the complete solution of linear inverse problems ?|date=November 2019}}, only one class of nonlinear inverse problems was so before 1970, that of inverse spectral and (one space dimension) [[inverse scattering problem]]s, after the seminal work of the Russian mathematical school ([[Mark Grigoryevich Krein|Krein]], [[Israel Gelfand|Gelfand]], Levitan, [[Vladimir Marchenko|Marchenko]]). A large review of the results has been given by Chadan and Sabatier in their book "Inverse Problems of Quantum Scattering Theory" (two editions in English, one in Russian).

In this kind of problem, data are properties of the spectrum of a linear operator which describe the scattering. The spectrum is made of [[eigenvalue]]s and [[eigenfunction]]s, forming together the "discrete spectrum", and generalizations, called the continuous spectrum. The very remarkable physical point is that scattering experiments give information only on the continuous spectrum, and that knowing its full spectrum is both necessary and sufficient in recovering the scattering operator. Hence we have invisible parameters, much more interesting than the null space which has a similar property in linear inverse problems. In addition, there are physical motions in which the spectrum of such an operator is conserved as a consequence of such motion. This phenomenon is governed by special nonlinear partial differential evolution equations, for example the [[Korteweg–de Vries equation]]. If the spectrum of the operator is reduced to one single eigenvalue, its corresponding motion is that of a single bump that propagates at constant velocity and without deformation, a solitary wave called a "[[soliton]]".

A perfect signal and its generalizations for the Korteweg–de Vries equation or other integrable nonlinear partial differential equations are of great interest, with many possible applications. This area has been studied as a branch of mathematical physics since the 1970s. Nonlinear inverse problems are also currently studied in many fields of applied science (acoustics, mechanics, quantum mechanics, electromagnetic scattering - in particular radar soundings, seismic soundings, and nearly all imaging modalities).

A final example related to the [[Riemann hypothesis]] was given by Wu and Sprung, the idea is that in the [[semiclassical physics|semiclassical]] [[old quantum theory]] the inverse of the potential inside the Hamiltonian is proportional to the [[half-derivative]] of the eigenvalues (energies) counting function&nbsp;''n''(''x'').