##
**Spectral and scattering theory for ordinary differential equations. Vol. I: Sturm-Liouville equations.**
*(English)*
Zbl 1468.34001

Universitext. Cham: Springer (ISBN 978-3-030-59087-1/pbk; 978-3-030-59088-8/ebook). ix, 379 p. (2020).

This excellent book is devoted to the spectral and scattering theory as well as the inverse theory for the Sturm-Liouville equation
\[
-(pu')'+ qu=wf,\tag{1}
\]
given on an arbitrary open interval \(l:=(0,\infty)\). To cope with the wealth of material this review starts with Chapter 4. The minimal requirements on the coefficients on the l.h.s. of (1) are \(1/p\), \(q\in L^1_{\text{loc}}(l)\), which were already introduced by M. R. Stone in his classic book [Linear transformations in Hilbert space and their applications to analysis. Providence, RI: American Mathematical Society (AMS) (1932; Zbl 0005.40003; JFM 58.0420.02)]. The assumptions on the weight function \(w\) the importance of which will soon become clear are \(w\in L^1_{\text{loc}}(l)\), \(w\ge 0\) a.e. and of course \(w\not\equiv 0\).

The maximal operator associated with (1) in \(L^2(l,w)\) is defined as the adjoint of a minimal relation \((u,f)\) given by (1) on functions with compact support. (Relations, operators – especially resolvents – and boundary-forms in a Hilbert space had already been introduced in Chapter 2.) The discussion of boundary conditions is given in more detail than usual; in particular, the role of symplectic matrices is emphasised. The limit-point (LP), limit-circle (LC) dichotomy is introduced by means of the rank of the boundary form that arises when (1) is integrated by parts. That this dichotomy is independent of the choice of the spectral parameter \(\lambda\in\mathbb{C}\) is deduced from a theorem by Derek Atkinson, Theorem 4.2.10. I knew this theorem, which is almost a necessary and sufficient LP criterion, but found this particular use surprising.

In Chapter 3, the authors had based their proof of the spectral theorem for a self-adjoint operator on the observation that the resolvent of such an operator is an analytic function which maps the upper half-plane into itself and so has a canonical Stieltjes representation that can be related to the abstract resolution of the identity. This procedure was probably first put forward by Doob and Koopman two years after and in response to Stone’s book [loc. cit]. In order to find an expansion theorem for a general self-adjoint realisation of (1) that is analogous to the Fourier transform (corresponding to \(p=w=1\) and \(q=0\) on \(l=(0,\infty)\); a bounded interval would give rise to a Fourier series), they consider resolvents again. However, they deviate again from the standard textbook procedure and separate the case where one resolvent suffices to describe the expansion theorem from the case where an essential spectrum of multiplicity two requires two resolvents. The chapter closes with a simple proof via Glazman’s decomposition principle of Molčanov’s famous necessary and sufficient condition for the discreteness of the spectrum of any self-adjoint realisation of (1) on \((0,\infty)\) with \(p=w=1\), \(q\) being bounded from below (Theorem 4.5.9).

When the r.h.s. of (1) can no longer provide the scalar product for a Hilbert space, one may hope for assistance from the Dirichlet integral \[ \int_I (p|u'|^2+ q|u|^2)\tag{2} \] which arises when (1) is integrated by parts. A spectral theory of this left-definite case is systematically presented here for the first time in a textbook. In order to encompass a physically significant case of the Camassa-Holm equation, viz., the breaking of water waves, it will be important to allow \(w\) to be a measure which changes sign (this is also why the Appendix contains, unusual for an ODE book, sections on Radon measures and Schwartz distributions).

Since nonnegativity of (2) for \(u\) in a sufficiently large set implies \(p>0\) a.e. (Theorem 4.5.2), \(p\) can be removed by a simple transformation and Chapter 5 starts with the assumptions \(p=1\), \(q\ge 0\) a.e. and \(w\in L^1_{\text{loc}}(l)\), \(w\not\equiv 0\). The underlying Hilbert space is therefore given by \[ H_1:=\{u\in \text{AC}_{\text{loc}}(l)\mid u',\,q^{1/2}u\in L^2(l)\}. \] Let \(L_c\) be the set of integrable functions with compact support in \(l\). Then it is shown that the Fréchet-Riesz representation theorem provides an integral operator \(G_0\) (symmetric on \(L_c\cap H_1\) with dense range in \(H_1\) with the help of which a symmetric relation \(T_1\) can be defined in \(H_1\). Its closure \(T_0\) is the minimal, its adjoint \(T_1\) the maximal relation associated with (1) in \(H_1\). (\(T_0\) is no longer an operator when \(w\) vanishes on a nonempty open set.) The rank of the boundary form connected with (1) and \(T_1\) is again either 2 (LC) or \(0\) (LP). Using a function \(W\in\text{AC}_{\text{loc}}(l)\) with \(W'=w\) a.e., very detailed descriptions of the four possible connections of LC and LP are given (Theorem 5.1.8). In contrast to the right-definite case the defect space for \(\lambda=0\), i.e., \((u,0)\in T_1\) for \(u\in H_1\), now has a special significance which is brought out in Theorems 5.1.9 and 5.1.13.

In case both endpoints are LC (or more generally under conditions on \(w\) or \(W\) that involve the diagonal of the kernel \(g_0\) of \(G_0\)) the resolvent of any self-adjoint realisation of (1) in \(H_1\) is compact (Theorem 5.2.1). In the general case the authors proceed as they had done in the right-definite case and construct one or two resolvents which have the canonical Stieltjes representation; here \(g_0\) plays again a decisive role. The final expansion theorems are Theorems 5.2.10 and 5.3.3.

The remaining four sections of Chapter 5 are devoted to obviating the restrictions on \(q\). It would have been helpful for the reader if it had been pointed out directly that the problem of establishing the nonnegativity of (2) occurred already in the calculus of variations and was solved by Jacobi by factorising the integrand with the help of a positive solution of (1) with \(f=\lambda u\). More generally, if this equation is non-oscillatory near one endpoint, then it has a canonical fundamental system of solutions, a principle solution (unique up to a factor) and a nonprinciple solution (cf. [P. Hartman, Ordinary differential equations. New York-London-Sydney: John Wiley and Sons, Inc. (1964; Zbl 0125.32102)]). The functions \(F_+\) and \(F_-\) from Definition 5.5.6 are nonprinciple solutions at the corresponding endpoints. \(u(x)/F_+(x)\) as \(x\to a\) means that \(u\) is a principle solution. It was Rellich who showed that precisely functions \(u\) in the domain of definition of the Friedrichs extension have this property (see, e.g., [F. Gesztesy et al., J. Differ. Equations 269, No. 9, 6448–6491 (2020; Zbl 1458.34143)]).

Chapter 6 starts by counting the number of zeros of solutions of (1) with \(f=\lambda u\) within a given interval by means of Prüfer transformations. Theorem 6.1.12 gives the contributions by Milne, Titchmarsh and Hartman its finishing touch. Then the behaviour of the solutions of (1) as well as that of the Weyl-Titchmarsh \(m\)-functions is discussed when \(\lambda\in\mathbb{C}\) becomes large, this both in the right- and in the left-definite case. The chapter concludes with the asymptotic behaviour of the kernel of the spectral function – not normally to be found in textbooks.

Chapter 7 investigates the extent to which the coefficients in (1) are determined by a prescribed \(m\)-function. A highlight is the proof of the affirmative Theorem 7.1.3 which elaborates the basically one-page proof C. Bennewitz presented in [Commun. Math. Phys. 218, No. 1, 131–132 (2001; Zbl 0982.34021)] 20 years ago. Here \(p=w=1\) on \(l=(0,b)\). A separated boundary condition is placed at the regular point \(0\); \(b\le\infty\) has to be supplemented with a boundary condition if necessary. The presence of \(p\) and \(w\) complicates the issue because of a possible unitary Liouville transformation; moreover, two endpoints of \(l\) that are singular may require two \(m\)-functions. However, all these situations are treated both in the right- and in the left-definite case.

The fruits of Chapters 5–7 are gathered in Chapter 8 where (1) with \(p=1\), \(f=\lambda u\) on \(\mathbb{R}\) is considered as a left-definite equation. Let \(q_0\in\mathbb{R}\) and \(q-q_0\), \(w-1\) and \(x|(q-q_0)w|\) be in \(L^1(\mathbb{R})\). The unique self-adjoint realisation \(T\) has \([q_0,\infty)\) as its absolutely continuous spectrum and its point spectrum consists of eigenvalues in \((-\infty,q_0)\) which are simple and can at most accumulate at \(-\infty\) (Theorem 8.3.1). Moreover, \(T\) has a fairly explicit spectral resolution in terms of Jost solutions (Theorem 8.3.7). However, as was just indicated, one can not expect that conversely the spectral information encoded in these functions determines both \(q\) and \(w\). Nevertheless, if \(q\) is also prescribed, then \(w\) is indeed determined (Corollary 8.4.1 to Theorem 8.4.9), and this is precisely the situation given with the Camassa-Holm equation where \(q=q_0=1/4\).

The book is written in a fresh and fluent style, always starting with informal explanations before proceeding to precisely formulated definitions and theorems. Proofs are well structured and detailed without being pedantic. Each chapter has notes sketching the historical development, pointing to other approaches or indicating wider aspects. Exercises are added to each paragraph and even to some of the appendices. It provides lecturers of courses on ODEs or functional analysis with new material to demonstrate to students the strength of classical real and complex analysis in conjunction with abstract methods. The researcher will be delighted to see a difficult topic slowly trickling down into graduate-level teaching.

The maximal operator associated with (1) in \(L^2(l,w)\) is defined as the adjoint of a minimal relation \((u,f)\) given by (1) on functions with compact support. (Relations, operators – especially resolvents – and boundary-forms in a Hilbert space had already been introduced in Chapter 2.) The discussion of boundary conditions is given in more detail than usual; in particular, the role of symplectic matrices is emphasised. The limit-point (LP), limit-circle (LC) dichotomy is introduced by means of the rank of the boundary form that arises when (1) is integrated by parts. That this dichotomy is independent of the choice of the spectral parameter \(\lambda\in\mathbb{C}\) is deduced from a theorem by Derek Atkinson, Theorem 4.2.10. I knew this theorem, which is almost a necessary and sufficient LP criterion, but found this particular use surprising.

In Chapter 3, the authors had based their proof of the spectral theorem for a self-adjoint operator on the observation that the resolvent of such an operator is an analytic function which maps the upper half-plane into itself and so has a canonical Stieltjes representation that can be related to the abstract resolution of the identity. This procedure was probably first put forward by Doob and Koopman two years after and in response to Stone’s book [loc. cit]. In order to find an expansion theorem for a general self-adjoint realisation of (1) that is analogous to the Fourier transform (corresponding to \(p=w=1\) and \(q=0\) on \(l=(0,\infty)\); a bounded interval would give rise to a Fourier series), they consider resolvents again. However, they deviate again from the standard textbook procedure and separate the case where one resolvent suffices to describe the expansion theorem from the case where an essential spectrum of multiplicity two requires two resolvents. The chapter closes with a simple proof via Glazman’s decomposition principle of Molčanov’s famous necessary and sufficient condition for the discreteness of the spectrum of any self-adjoint realisation of (1) on \((0,\infty)\) with \(p=w=1\), \(q\) being bounded from below (Theorem 4.5.9).

When the r.h.s. of (1) can no longer provide the scalar product for a Hilbert space, one may hope for assistance from the Dirichlet integral \[ \int_I (p|u'|^2+ q|u|^2)\tag{2} \] which arises when (1) is integrated by parts. A spectral theory of this left-definite case is systematically presented here for the first time in a textbook. In order to encompass a physically significant case of the Camassa-Holm equation, viz., the breaking of water waves, it will be important to allow \(w\) to be a measure which changes sign (this is also why the Appendix contains, unusual for an ODE book, sections on Radon measures and Schwartz distributions).

Since nonnegativity of (2) for \(u\) in a sufficiently large set implies \(p>0\) a.e. (Theorem 4.5.2), \(p\) can be removed by a simple transformation and Chapter 5 starts with the assumptions \(p=1\), \(q\ge 0\) a.e. and \(w\in L^1_{\text{loc}}(l)\), \(w\not\equiv 0\). The underlying Hilbert space is therefore given by \[ H_1:=\{u\in \text{AC}_{\text{loc}}(l)\mid u',\,q^{1/2}u\in L^2(l)\}. \] Let \(L_c\) be the set of integrable functions with compact support in \(l\). Then it is shown that the Fréchet-Riesz representation theorem provides an integral operator \(G_0\) (symmetric on \(L_c\cap H_1\) with dense range in \(H_1\) with the help of which a symmetric relation \(T_1\) can be defined in \(H_1\). Its closure \(T_0\) is the minimal, its adjoint \(T_1\) the maximal relation associated with (1) in \(H_1\). (\(T_0\) is no longer an operator when \(w\) vanishes on a nonempty open set.) The rank of the boundary form connected with (1) and \(T_1\) is again either 2 (LC) or \(0\) (LP). Using a function \(W\in\text{AC}_{\text{loc}}(l)\) with \(W'=w\) a.e., very detailed descriptions of the four possible connections of LC and LP are given (Theorem 5.1.8). In contrast to the right-definite case the defect space for \(\lambda=0\), i.e., \((u,0)\in T_1\) for \(u\in H_1\), now has a special significance which is brought out in Theorems 5.1.9 and 5.1.13.

In case both endpoints are LC (or more generally under conditions on \(w\) or \(W\) that involve the diagonal of the kernel \(g_0\) of \(G_0\)) the resolvent of any self-adjoint realisation of (1) in \(H_1\) is compact (Theorem 5.2.1). In the general case the authors proceed as they had done in the right-definite case and construct one or two resolvents which have the canonical Stieltjes representation; here \(g_0\) plays again a decisive role. The final expansion theorems are Theorems 5.2.10 and 5.3.3.

The remaining four sections of Chapter 5 are devoted to obviating the restrictions on \(q\). It would have been helpful for the reader if it had been pointed out directly that the problem of establishing the nonnegativity of (2) occurred already in the calculus of variations and was solved by Jacobi by factorising the integrand with the help of a positive solution of (1) with \(f=\lambda u\). More generally, if this equation is non-oscillatory near one endpoint, then it has a canonical fundamental system of solutions, a principle solution (unique up to a factor) and a nonprinciple solution (cf. [P. Hartman, Ordinary differential equations. New York-London-Sydney: John Wiley and Sons, Inc. (1964; Zbl 0125.32102)]). The functions \(F_+\) and \(F_-\) from Definition 5.5.6 are nonprinciple solutions at the corresponding endpoints. \(u(x)/F_+(x)\) as \(x\to a\) means that \(u\) is a principle solution. It was Rellich who showed that precisely functions \(u\) in the domain of definition of the Friedrichs extension have this property (see, e.g., [F. Gesztesy et al., J. Differ. Equations 269, No. 9, 6448–6491 (2020; Zbl 1458.34143)]).

Chapter 6 starts by counting the number of zeros of solutions of (1) with \(f=\lambda u\) within a given interval by means of Prüfer transformations. Theorem 6.1.12 gives the contributions by Milne, Titchmarsh and Hartman its finishing touch. Then the behaviour of the solutions of (1) as well as that of the Weyl-Titchmarsh \(m\)-functions is discussed when \(\lambda\in\mathbb{C}\) becomes large, this both in the right- and in the left-definite case. The chapter concludes with the asymptotic behaviour of the kernel of the spectral function – not normally to be found in textbooks.

Chapter 7 investigates the extent to which the coefficients in (1) are determined by a prescribed \(m\)-function. A highlight is the proof of the affirmative Theorem 7.1.3 which elaborates the basically one-page proof C. Bennewitz presented in [Commun. Math. Phys. 218, No. 1, 131–132 (2001; Zbl 0982.34021)] 20 years ago. Here \(p=w=1\) on \(l=(0,b)\). A separated boundary condition is placed at the regular point \(0\); \(b\le\infty\) has to be supplemented with a boundary condition if necessary. The presence of \(p\) and \(w\) complicates the issue because of a possible unitary Liouville transformation; moreover, two endpoints of \(l\) that are singular may require two \(m\)-functions. However, all these situations are treated both in the right- and in the left-definite case.

The fruits of Chapters 5–7 are gathered in Chapter 8 where (1) with \(p=1\), \(f=\lambda u\) on \(\mathbb{R}\) is considered as a left-definite equation. Let \(q_0\in\mathbb{R}\) and \(q-q_0\), \(w-1\) and \(x|(q-q_0)w|\) be in \(L^1(\mathbb{R})\). The unique self-adjoint realisation \(T\) has \([q_0,\infty)\) as its absolutely continuous spectrum and its point spectrum consists of eigenvalues in \((-\infty,q_0)\) which are simple and can at most accumulate at \(-\infty\) (Theorem 8.3.1). Moreover, \(T\) has a fairly explicit spectral resolution in terms of Jost solutions (Theorem 8.3.7). However, as was just indicated, one can not expect that conversely the spectral information encoded in these functions determines both \(q\) and \(w\). Nevertheless, if \(q\) is also prescribed, then \(w\) is indeed determined (Corollary 8.4.1 to Theorem 8.4.9), and this is precisely the situation given with the Camassa-Holm equation where \(q=q_0=1/4\).

The book is written in a fresh and fluent style, always starting with informal explanations before proceeding to precisely formulated definitions and theorems. Proofs are well structured and detailed without being pedantic. Each chapter has notes sketching the historical development, pointing to other approaches or indicating wider aspects. Exercises are added to each paragraph and even to some of the appendices. It provides lecturers of courses on ODEs or functional analysis with new material to demonstrate to students the strength of classical real and complex analysis in conjunction with abstract methods. The researcher will be delighted to see a difficult topic slowly trickling down into graduate-level teaching.

Reviewer: Hubert Kalf (München)

### MSC:

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

34L25 | Scattering theory, inverse scattering involving ordinary differential operators |

34A55 | Inverse problems involving ordinary differential equations |

34B24 | Sturm-Liouville theory |

34L05 | General spectral theory of ordinary differential operators |

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