This book presents the main results and methods on inverse spectral problems for Sturm-Liouville differential operators and their applications. Inverse problems of spectral analysis consist in recovering operators from their spectral characteristics. Such problems often appear
in mathematics, mechanics, physics, electronics, geophysics, meteorology and other
branches of natural sciences. Inverse problems also play an important role in solving nonlinear evolution equations in mathematical physics. Interest in this subject has been increasing permanently because of the appearance of new important applications, and nowadays the
inverse problem theory develops intensively all over the world.
The greatest success in spectral theory in general, and in particular in inverse spectral problems has been achieved for the Sturm-Liouville operator
ℓy := −y′′ + q(x)y, (1)
which also is called the one-dimensional Schr¨odinger operator. The first studies on the
spectral theory of such operators were performed by D. Bernoulli, J. d‘Alembert, L. Euler, J. Liouville and Ch. Sturm in connection with the solution of the equation describing the
vibration of a string. An intensive development of the spectral theory for various classes of differential and integral operators and for operators in abstract spaces took place in the
XX-th century. Deep ideas here are due to G. Birkhoff, D. Hilbert, J. von Neumann, V.
Steklov, M. Stone, H. Weyl and many other mathematicians. The main results on inverse spectral problems appear in the second half of the XX-th century. We mention here the works by R. Beals, G. Borg, L.D. Faddeev, M.G. Gasymov, I.M. Gelfand, B.M. Levitan, I.G.
Khachatryan, M.G. Krein, N. Levinson, Z.L. Leibenson, V.A. Marchenko, L.A. Sakhnovich, E. Trubowitz, V.A. Yurko and others (see Section 1.9 for details). An important role in the inverse spectral theory for the Sturm-Liouville operator was played by the transformation operator method. But this method turned out to be unsuitable for many important classes of inverse problems being more complicated than the Sturm-Liouville operator. At present time other effective methods for solving inverse spectral problems have been created. Among them we point out the method of spectral mappings connected with ideas of the contour integral method. This method seems to be perspective for inverse spectral problems. The created methods allowed to solve a number of important problems in various branches of natural sciences.
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In recent years there appeared new areas for applications of inverse spectral problems. We mention a remarkable method for solving some nonlinear evolution equations of mathematical physics connected with the use of inverse spectral problems. Another important class of inverse problems, which often
appear in applications, is the inverse problem of recovering differential equations from incomplete spectral information when only a part of the spectral information is available for measurement. Many applications are connected with inverse problems for differential equations having singularities and turning points, for higher-order differential operators, for differential operators with delay or other types of ”aftereffect”.
The main goals of this book are as follows:
• To present a fairly elementary and complete introduction to the inverse problem theory for ordinary differential equations which is suitable for the ”first reading” and accessible not only to mathematicians but also to physicists, engineers and students. Note that
the book requires knowledge only of classical analysis and the theory of ordinary linear differential equations.
• To describe the main ideas and methods in the inverse problem theory using the Sturm- Liouville operator as a model. Up to now many of these ideas, in particular those which appeared in recent years, are presented in journals only. It is very important that the
methods, provided in this book, can be used (and have been already used) not only for Sturm-Liouville operators, but also for solving inverse problems for other more complicated classes of operators such as differential operators of arbitrary orders, differential operators with singularities and turning points, pencils of operators, integro-differential and integral operators and others.
• To reflect various applications of the inverse spectral problems in natural sciences and engineering.
The book is organized as follows. In Chapter 1, Sturm-Liouville operators (1) on a finite interval are considered. In Sections 1.1-1.3 we study properties of spectral characteristics and
eigenfunctions, and prove a completeness theorem and an expansion theorem. Sections 1.4- 1.8 are devoted to the inverse problem theory. We prove uniqueness theorems, give algorithms
for the solution of the inverse problems considered, provide necessary and sufficient conditions for their solvability, and study stability of the solutions. We present several methods for solving inverse problems. The transformation operator method, in which the inverse problem is reduced to the solution of a linear integral equation, is described in Section 1.5. In Section 1.6 we present the method of spectral mappings, in which ideas of the contour integral method are used. The central role there is played by the so-called main equation of the inverse problem which is a linear equation in a corresponding Banach space. We give a derivation of the main equation, prove its unique solvability and provide explicit formulae for the solution
of the inverse problem. At present time the contour integral method seems to be the most universal tool in the inverse problem theory for ordinary differential operators. It has a wide area for applications in various classes of inverse problems.
In the method of standard models, which is described in Section 1.7, a sequence of model differential operators approximating the unknown operator are constructed. In Section 1.8 we provide the method for the local solution of the inverse problem from two spectra which is due to G. Borg [Bor1]. In this method, the inverse
problem is reduced to a special nonlinear integral equation, which can be solved locally.
Chapter 2 is devoted to Sturm-Liouville operators on the half-line. First we consider nonselfadjoint
operators with integrable potentials. Using the contour integral method we study
the inverse problem of recovering the Sturm-Liouville operator from its Weyl function. Then locally integrable complex-valued potentials are studied, and the inverse problem is solved by specifying the generalized Weyl function. In Chapter 3 Sturm-Liouville operators on the line are considered, and the inverse scattering problem is studied. In Chapter 4 we provide a number of applications of the inverse problem theory in natural sciences and engineering:
we consider the solution of the Korteweg-de Vries equation on the line and on the half-line, solve the inverse problem of constructing parameters of a medium from incomplete spectral information, and study boundary value problems with aftereffect, inverse problems in elasticity theory and others.
There exists an extensive literature devoted to inverse spectral problems. Instead of trying to give a complete list of all relevant contributions, we mention only monographs, survey articles and the most important papers, and refer to the references therein. In Section 1.9 we give a short review on literature on the inverse problem theory.
The full text can be found as pdf in the internet (search for: Inverse Sturm–Liouville Problems and Their Applications)