##
**Inverse spectral theory.**
*(English)*
Zbl 0623.34001

The harmless looking Sturm-Liouville equation
\[
(SL)\quad - y''+q(x)y=\lambda y\quad (0\leq x\leq 1)
\]
exhibits a lot of interesting and fascinating phenomena. Some information on the ”interaction” between the analytical properties of the potential q and the structure of the spectrum of equation (SL) may be found, for example, in the book ”Hill’s equation” by W. Magnus and S. Winkler [(1966; Zbl 0158.096)]. After G. Borg’s pioneering paper ”Eine Umkehrung der Sturm- Liouvilleschen Eigenwertaufgabe” [Acta Math. 78, 1–96 (1946; Zbl 0063.00523)] the inverse spectral problem for equation (SL) has found particular attention and was essentially developed by A. A. Androshchuk, V. I. Aratyunyan, V. Barcilon, P. Deift, G. Eskin, I. M. Gel’fand, O. Hald, H. Hochstadt, E. L. Isaacson, N. Levinson, B. M. Levitan, V. I. Marchenko, H. P. McKean, J. R. McLaughlin, P. Sabatier, J. Sylvester, E. Trubowitz, E. Uhlmann, and others. Many important results, however, are scattered in the research literature, and have never appeared in book form.

In the present book, the authors make the attempt to provide a self- contained thorough study of both the direct and inverse spectral problem for equation (SL), subject to Dirichlet boundary conditions (1) \(y(0)=y(1)=0\), where the potential is assumed throughout to belong to the space \(L^ 2(0,1).\)

The contents of the book may be described as follows. In Chapter 1, the fundamental solutions \(y_ 1\) and \(y_ 2\) of (SL) satisfying \(y_ 1(0)=y_ 2'(0)=1\) and \(y_ 1'(0)=y_ 2(0)=0\) are constructed and analyzed, mainly in view of their asymptotic behaviour with respect to \(\lambda\in {\mathbb{C}}\) and \(q\in L^ 2\). The Dirichlet problem for (SL) is studied in detail in Chapter 2; in particular, it is shown that the spectrum consists of a monotonically increasing sequence of simple eigenvalues \(\mu_ n=\mu_ n(q)\) with asymptotic behaviour (2) \(\mu_ n=n^ 2\pi^ 2+const+{\tilde \mu}_ n,\) where \({\tilde \mu}_ n\) is a ”remainder” sequence in \(\ell^ 2\). Moreover, a crucial point is that the corresponding eigenfunctions \(g_ n=g_ n(x,q)\) satisfy the ”orthogonality relation” (3) \(<g^ 2_ n,(d/dx)g^ 2_ n>=0;\) similar relations hold for products of the ”cosine-like” solutions \(y_ 1=y_ 1(x,\mu_ n)\) and ”sine-like” solutions \(y_ 2=y_ 2(x,\mu_ n)\) (see above).

The main part of the book starts with Chapter 3 and deals with the inverse Dirichlet problem for (SL). Here the main results can be summarized as follows. If the potential q is even (i.e. \(q(1-x)=q(x))\), then q is uniquely determined by the corresponding eigenvalues \(\mu_ n\). If q is an arbitrary \(L^ 2\) potential, then q is uniquely determined by the eigenvalues \(\mu_ n\) and the ”normalizing constants” \[ (4)\quad \kappa_ n=\kappa_ n(q)=\log (| g_ n'(1,q)| /| g_ n'(0,q)|); \] in particular, \(\kappa_ n(q)\equiv 0\) iff q is even. Given any increasing sequence \(\mu_ n\) of real numbers with asymptotic behaviour (2), one can find an \(L^ 2\) potential q such that \(\mu_ n=\mu_ n(q)\). Further, given a fixed potential \(p\in L^ 2\), the ”isospectral set” \[ (5)\quad M(p)=\{q: q\in L^ 2,\quad \mu_ n(q)=\mu_ n(p),\quad n=1,2,...\} \] is an unbounded connected real- analytic submanifold of \(L^ 2\) which is diffeomorphic to the linear space \(h^ 1\) of all sequences \(\kappa_ n\) with \(\sum n^ 2\kappa^ 2_ n<\infty\). More precisely, the normalizing constants (4) form a global coordinate system on each M(p). Every isospectral set contains exactly one even function which is at the same time the closest point to the origin of \(L^ 2\); similarly, every isospectral set contains exactly one function which vanishes on [0,1/2] (or [1/2,1]). By considering vector fields and exponential maps on M(p), it is possible (at least theoretically) to construct the solution of the inverse problem explicitly.

The book is well written and carefully printed (the only misprint the reviewer could find is on p. 47, 3rd line from the bottom: \(\ell^ 2_ 1\times {\mathbb{R}}\times \ell^ 2)\), and contains a wealth of interesting results, illuminating discussions, and exercises of various difficulty. Obviously, most results presented are not new, but the approach is novel. It is in fact the authors’ merit to use the elegant and ”geometrical” language of elementary global analysis, rather than the somewhat cumbersome Gel’fand-Levitan technique. From this point of view, the book is useful and accessible to a wide mathematical audience.

An essential flaw of the book, however, concerns the References. Taking into account the general title, the authors should have also mentioned other techniques which are different (sometimes equivalent), but equally interesting. For instance, the classical Gel’fand-Levitan approach uses the so-called ”spectral function” [see, e.g., I. M. Gel’fand and B. M. Levitan, Izv. Akad. Nauk SSSR, Ser. Mat. 15, 309-360 (1951; Zbl 0044.093)] which determines the potential uniquely. In the case of the Dirichlet problem (1), this function is a decreasing step function with jumping discontinuities at each \(\mu_ n\) of height \(\| g_ n(\cdot,q)\|\); by the well-known identity \[ (6)\quad \| g_ n(\cdot,q)\|^ 2=(\partial /\partial x)g_ n(1,q)(\partial /\partial \lambda)g_ n(1,q), \] this is equivalent to knowing the sequences \(\mu_ n\) and \(\kappa_ n\). Another way is to consider the problem (SL) over the ”variable” interval [0,t] and to study the eigenvalues \(\mu_ n=\mu_ n(q,t)\) as functions of t. In this case, the knowledge of both \(\mu_ n(q,1)\) and \(\mu_ n'(q,1)\) allows one to identify the potential q [see W. A. Pranger, ”Derivatives of the eigenvalues in the inverse Sturm-Liouville problem” (to appear)]; since \[ (7)\quad | \frac{\partial}{\partial t}\mu_ n(q,1)| \| g_ n(\cdot,q)\|^ 2=\exp \kappa_ n(q), \] this is again equivalent to knowing both \(\mu_ n\) and \(\kappa_ n\). Finally, in a recent generalization to higher dimensions [see A. Nachman, J. Sylvester and G. Uhlmann, ”An n-dimensional Borg-Levinson theorem” (to appear)], the authors consider the so-called ”Dirichlet-Neumann map” which allows them to obtain a uniqueness result for the (smooth) potential q in the Schrödinger equation \[ (8)\quad -\Delta u+q(x)u=\lambda u\quad (x\in \Omega), \] subject to Dirichlet boundary conditions.

The book is certainly a valuable contribution to a continuously growing interesting field of contemporary mathematical analysis, and will certainly become a standard reference. In the Preface, the authors announce a second volume which will be concerned with the ”periodic” problem (which in some sense is more interesting than the Dirichlet problem and gives rise to many new features). Moreover, a forthcoming survey on (SL) with either general ”separated” boundary conditions \[ (9)\quad y(0) \cos \alpha +y'(0) \sin \alpha =0,\quad y(1) \cos \beta +y'(1) \sin \beta =0\quad (0\leq \alpha,\beta <\pi) \] or general ”periodic” boundary conditions \[ (10)\quad a y(1)+b y'(1)=y(0),\quad c y(1)+d y'(1)=y'(0),\quad (ad-bc=1) \] by J. Ralston and E. Trubowitz will be published in the J. Dyn. Syst. Meas. Control.

In the present book, the authors make the attempt to provide a self- contained thorough study of both the direct and inverse spectral problem for equation (SL), subject to Dirichlet boundary conditions (1) \(y(0)=y(1)=0\), where the potential is assumed throughout to belong to the space \(L^ 2(0,1).\)

The contents of the book may be described as follows. In Chapter 1, the fundamental solutions \(y_ 1\) and \(y_ 2\) of (SL) satisfying \(y_ 1(0)=y_ 2'(0)=1\) and \(y_ 1'(0)=y_ 2(0)=0\) are constructed and analyzed, mainly in view of their asymptotic behaviour with respect to \(\lambda\in {\mathbb{C}}\) and \(q\in L^ 2\). The Dirichlet problem for (SL) is studied in detail in Chapter 2; in particular, it is shown that the spectrum consists of a monotonically increasing sequence of simple eigenvalues \(\mu_ n=\mu_ n(q)\) with asymptotic behaviour (2) \(\mu_ n=n^ 2\pi^ 2+const+{\tilde \mu}_ n,\) where \({\tilde \mu}_ n\) is a ”remainder” sequence in \(\ell^ 2\). Moreover, a crucial point is that the corresponding eigenfunctions \(g_ n=g_ n(x,q)\) satisfy the ”orthogonality relation” (3) \(<g^ 2_ n,(d/dx)g^ 2_ n>=0;\) similar relations hold for products of the ”cosine-like” solutions \(y_ 1=y_ 1(x,\mu_ n)\) and ”sine-like” solutions \(y_ 2=y_ 2(x,\mu_ n)\) (see above).

The main part of the book starts with Chapter 3 and deals with the inverse Dirichlet problem for (SL). Here the main results can be summarized as follows. If the potential q is even (i.e. \(q(1-x)=q(x))\), then q is uniquely determined by the corresponding eigenvalues \(\mu_ n\). If q is an arbitrary \(L^ 2\) potential, then q is uniquely determined by the eigenvalues \(\mu_ n\) and the ”normalizing constants” \[ (4)\quad \kappa_ n=\kappa_ n(q)=\log (| g_ n'(1,q)| /| g_ n'(0,q)|); \] in particular, \(\kappa_ n(q)\equiv 0\) iff q is even. Given any increasing sequence \(\mu_ n\) of real numbers with asymptotic behaviour (2), one can find an \(L^ 2\) potential q such that \(\mu_ n=\mu_ n(q)\). Further, given a fixed potential \(p\in L^ 2\), the ”isospectral set” \[ (5)\quad M(p)=\{q: q\in L^ 2,\quad \mu_ n(q)=\mu_ n(p),\quad n=1,2,...\} \] is an unbounded connected real- analytic submanifold of \(L^ 2\) which is diffeomorphic to the linear space \(h^ 1\) of all sequences \(\kappa_ n\) with \(\sum n^ 2\kappa^ 2_ n<\infty\). More precisely, the normalizing constants (4) form a global coordinate system on each M(p). Every isospectral set contains exactly one even function which is at the same time the closest point to the origin of \(L^ 2\); similarly, every isospectral set contains exactly one function which vanishes on [0,1/2] (or [1/2,1]). By considering vector fields and exponential maps on M(p), it is possible (at least theoretically) to construct the solution of the inverse problem explicitly.

The book is well written and carefully printed (the only misprint the reviewer could find is on p. 47, 3rd line from the bottom: \(\ell^ 2_ 1\times {\mathbb{R}}\times \ell^ 2)\), and contains a wealth of interesting results, illuminating discussions, and exercises of various difficulty. Obviously, most results presented are not new, but the approach is novel. It is in fact the authors’ merit to use the elegant and ”geometrical” language of elementary global analysis, rather than the somewhat cumbersome Gel’fand-Levitan technique. From this point of view, the book is useful and accessible to a wide mathematical audience.

An essential flaw of the book, however, concerns the References. Taking into account the general title, the authors should have also mentioned other techniques which are different (sometimes equivalent), but equally interesting. For instance, the classical Gel’fand-Levitan approach uses the so-called ”spectral function” [see, e.g., I. M. Gel’fand and B. M. Levitan, Izv. Akad. Nauk SSSR, Ser. Mat. 15, 309-360 (1951; Zbl 0044.093)] which determines the potential uniquely. In the case of the Dirichlet problem (1), this function is a decreasing step function with jumping discontinuities at each \(\mu_ n\) of height \(\| g_ n(\cdot,q)\|\); by the well-known identity \[ (6)\quad \| g_ n(\cdot,q)\|^ 2=(\partial /\partial x)g_ n(1,q)(\partial /\partial \lambda)g_ n(1,q), \] this is equivalent to knowing the sequences \(\mu_ n\) and \(\kappa_ n\). Another way is to consider the problem (SL) over the ”variable” interval [0,t] and to study the eigenvalues \(\mu_ n=\mu_ n(q,t)\) as functions of t. In this case, the knowledge of both \(\mu_ n(q,1)\) and \(\mu_ n'(q,1)\) allows one to identify the potential q [see W. A. Pranger, ”Derivatives of the eigenvalues in the inverse Sturm-Liouville problem” (to appear)]; since \[ (7)\quad | \frac{\partial}{\partial t}\mu_ n(q,1)| \| g_ n(\cdot,q)\|^ 2=\exp \kappa_ n(q), \] this is again equivalent to knowing both \(\mu_ n\) and \(\kappa_ n\). Finally, in a recent generalization to higher dimensions [see A. Nachman, J. Sylvester and G. Uhlmann, ”An n-dimensional Borg-Levinson theorem” (to appear)], the authors consider the so-called ”Dirichlet-Neumann map” which allows them to obtain a uniqueness result for the (smooth) potential q in the Schrödinger equation \[ (8)\quad -\Delta u+q(x)u=\lambda u\quad (x\in \Omega), \] subject to Dirichlet boundary conditions.

The book is certainly a valuable contribution to a continuously growing interesting field of contemporary mathematical analysis, and will certainly become a standard reference. In the Preface, the authors announce a second volume which will be concerned with the ”periodic” problem (which in some sense is more interesting than the Dirichlet problem and gives rise to many new features). Moreover, a forthcoming survey on (SL) with either general ”separated” boundary conditions \[ (9)\quad y(0) \cos \alpha +y'(0) \sin \alpha =0,\quad y(1) \cos \beta +y'(1) \sin \beta =0\quad (0\leq \alpha,\beta <\pi) \] or general ”periodic” boundary conditions \[ (10)\quad a y(1)+b y'(1)=y(0),\quad c y(1)+d y'(1)=y'(0),\quad (ad-bc=1) \] by J. Ralston and E. Trubowitz will be published in the J. Dyn. Syst. Meas. Control.

Reviewer: J.Appell

### MSC:

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |

34L99 | Ordinary differential operators |

34B20 | Weyl theory and its generalizations for ordinary differential equations |

34C30 | Manifolds of solutions of ODE (MSC2000) |

34A55 | Inverse problems involving ordinary differential equations |