##
**Boundary value problems, Weyl functions, and differential operators.**
*(English)*
Zbl 1457.47001

Monographs in Mathematics 108. Cham: Birkhäuser (ISBN 978-3-030-36713-8/hbk; 978-3-030-36716-9/pbk; 978-3-030-36714-5/ebook). vii, 772 p., open access (2020).

The monograph under review presents a comprehensive overview of boundary value problems in an operator theoretic setting with applications to differential equations. The book is advertising the concept of boundary triplets and Weyl functions from spectral theory of symmetric and selfadjoint operators in Hilbert spaces, which is widely used by mathematicians working in operator theory and applications. In fact,
this monograph closes a certain gap, since a proper source and reference text for the abstract theory of boundary triplet techniques has not yet existed in the mathematical literature. The authors have written a self-contained and easily accessible text, and hence this monograph can also be seen as an introduction to this topic and its various applications. It will also be useful for researchers in other related areas and graduate students.

The book starts with a nice example, the Sturm-Liouville differential expression \(L := - \frac{d^2}{dx^2} + V\) with a bounded real potential \(V\) on the half line \({\mathbb R}^+ := (0, \infty)\). It is clear that the associated maximal operator in the Hilbert space \(L^2({\mathbb R}^+)\) with domain \({\mathfrak D}_{\max}\) (where the expression \(Lf \in L^2({\mathbb R}^+)\) makes sense) is the adjoint \(S^*\) of the symmetric (minimal) operator \(S\) defined by \(Sf := -f'' + V f\) for all \(f \in {\mathfrak D}_{\max}\) with \(f(0) = f'(0) = 0\). This domain can be rewritten as \(\operatorname{dom} S = \ker \Gamma_0 \cap \ker \Gamma_1\) with the boundary mappings \(\Gamma_0, \Gamma_1: \operatorname{dom} S^* \to {\mathbb C}\) given by \(\Gamma_0f := f(0), \, \Gamma_1f := f'(0)\). These mappings satisfy the “abstract Green identity” \((S^*f,g)_{L^2({\mathbb R}^+)} - (f,S^*g)_{ L^2({\mathbb R}^+)} = (\Gamma_1f,\Gamma_0g)_{\mathbb C}-(\Gamma_0f,\Gamma_1g)_{\mathbb C}\) (where \((\cdot,\cdot)_{\mathbb C}\) denotes the scalar product in the Hilbert space \({\mathbb C}\)), which implies that \(({\mathbb C}, \, \Gamma_0, \, \Gamma_1)\) is a boundary triplet. It allows, for example, the characterization of all self-adjoint extensions of \(S\) via \(A_\tau f := -f'' + V f\) with \(\operatorname{dom} A_\tau := \ker (\Gamma_1 - \tau \Gamma_0)\), where \(\tau\) runs though \({\mathbb R} \cup \{\infty\}\). (Here, for \(\tau = \infty\), this means that \(\operatorname{dom} A_\infty = \ker \Gamma_0\).) In other words, we are talking about the self-adjoint boundary conditions \(f'(0) = \tau f(0)\) for \(\tau \in {\mathbb R}\) and \(f(0) = 0\) for \(\tau = \infty\).

It is well known that the spectrum of \(A_\infty\) can be characterized by means of the Titchmarsh-Weyl \(m\)-function. This function \(m\) is determined by the unique property that, for \(\lambda \in {\mathbb C} \setminus {\mathbb R}\), the function \(\varphi_\lambda + m(\lambda) \psi_\lambda\) belongs to \(L^2({\mathbb R}^+)\), where \(\varphi_\lambda, \psi_\lambda\) form a fundamental system of \(-f'' + V f = \lambda f\) with \(\varphi_\lambda (0) = \psi_\lambda'(0) = 1\), \(\varphi_\lambda' (0) = \psi_\lambda(0) = 0\). In a semibounded (or even non-negative) setting, a different approach is given by the sesquilinear forms \({\mathfrak t}[f,g] := \int_0^\infty (f'\bar{g}' + V f\bar{g}) \, dx\) and \({\mathfrak t}_\tau[f,g] := {\mathfrak t}[f,g] + \tau f(0)\overline{g(0)}\) on suitable domains. This form approach leads to the study of the Friedrichs and the Krein-von Neumann extension of \(S\). Here, the so-called boundary pair \(({\mathbb C}, \, \Lambda)\) with the mapping \(\Lambda: \operatorname{dom} {\mathfrak t} \rightarrow {\mathbb C}\), \(\Lambda f := f(0)\), establishes a connection to the above boundary triplet. Now, the abstract theory of the monograph can be regarded more or less as a generalization of these considerations (and a lot more) to the setting of an arbitrary symmetric operator \(S\) (or, even more generally, relation \(S\)) with equal defect numbers in a Hilbert space \({\mathfrak H}\).

The first part of the monograph consists of Chapters 1–5, presenting the abstract theory. Chapter 1 is in some sense is preparatory, but it also contains a lot of material on linear operators and relations which goes beyond the “standard knowledge”, e.g., decomposition of linear relations, graph and resolvent convergence of operators and relations, parametric representations, and Nevanlinna families. All this material is used later in the text. Chapter 2 goes to the heart of the matter, here boundary triplets, Weyl functions and their properties are studied. An essential point in this chapter is Krein’s formula in a very general context. Chapter 3 continues the theme of Chapter 2; here, the analytic properties of the Weyl function are related to the resolvent and the spectrum of a self-adjoint extension. Chapter 4 is about a closely related inverse problem: a given Nevanlinna function is interpreted as a Weyl function of a boundary triplet. Corresponding models are often constructed with the help of reproducing kernel Hilbert spaces, a topic that is also discussed in Chapter 4. Finally, Chapter 5 complements the earlier chapters by discussing semibounded extensions of semibounded operators or relations in detail. Here, the key feature is the notion of a boundary pair, which connects boundary triplet and form techniques. Much of this material seems to be new in the mathematical literature.

The second part of the monograph is of a more applied nature. In Chapters 6–8, the authors show how the abstract methods fit with concrete boundary value problems. The classical and standard example of a Sturm-Liouville operator (as indicated above) is discussed in Chapter 6 in full detail. Of particular interest is the interplay with boundary pairs, boundary triplets and form techniques in this chapter. The next chapter treats a class of first order systems of differential equations. Here, it becomes clear that multivalued operators, i.e., linear relations, arise naturally in applications. Finally, Chapter 8 studies Schrödinger operators on bounded domains. A complete systematic description of all proper boundary conditions that lead to self-adjoint (or maximal dissipative/accumulative) realizations of an elliptic PDE (and their spectral analysis) is an interesting topic. The boundary triplet method is obviously most suitable and shows its full power here. The monograph also contains some Appendices to keep the text self-contained, an impressive List of References, and some Notes and interesting Historical Comments.

The book starts with a nice example, the Sturm-Liouville differential expression \(L := - \frac{d^2}{dx^2} + V\) with a bounded real potential \(V\) on the half line \({\mathbb R}^+ := (0, \infty)\). It is clear that the associated maximal operator in the Hilbert space \(L^2({\mathbb R}^+)\) with domain \({\mathfrak D}_{\max}\) (where the expression \(Lf \in L^2({\mathbb R}^+)\) makes sense) is the adjoint \(S^*\) of the symmetric (minimal) operator \(S\) defined by \(Sf := -f'' + V f\) for all \(f \in {\mathfrak D}_{\max}\) with \(f(0) = f'(0) = 0\). This domain can be rewritten as \(\operatorname{dom} S = \ker \Gamma_0 \cap \ker \Gamma_1\) with the boundary mappings \(\Gamma_0, \Gamma_1: \operatorname{dom} S^* \to {\mathbb C}\) given by \(\Gamma_0f := f(0), \, \Gamma_1f := f'(0)\). These mappings satisfy the “abstract Green identity” \((S^*f,g)_{L^2({\mathbb R}^+)} - (f,S^*g)_{ L^2({\mathbb R}^+)} = (\Gamma_1f,\Gamma_0g)_{\mathbb C}-(\Gamma_0f,\Gamma_1g)_{\mathbb C}\) (where \((\cdot,\cdot)_{\mathbb C}\) denotes the scalar product in the Hilbert space \({\mathbb C}\)), which implies that \(({\mathbb C}, \, \Gamma_0, \, \Gamma_1)\) is a boundary triplet. It allows, for example, the characterization of all self-adjoint extensions of \(S\) via \(A_\tau f := -f'' + V f\) with \(\operatorname{dom} A_\tau := \ker (\Gamma_1 - \tau \Gamma_0)\), where \(\tau\) runs though \({\mathbb R} \cup \{\infty\}\). (Here, for \(\tau = \infty\), this means that \(\operatorname{dom} A_\infty = \ker \Gamma_0\).) In other words, we are talking about the self-adjoint boundary conditions \(f'(0) = \tau f(0)\) for \(\tau \in {\mathbb R}\) and \(f(0) = 0\) for \(\tau = \infty\).

It is well known that the spectrum of \(A_\infty\) can be characterized by means of the Titchmarsh-Weyl \(m\)-function. This function \(m\) is determined by the unique property that, for \(\lambda \in {\mathbb C} \setminus {\mathbb R}\), the function \(\varphi_\lambda + m(\lambda) \psi_\lambda\) belongs to \(L^2({\mathbb R}^+)\), where \(\varphi_\lambda, \psi_\lambda\) form a fundamental system of \(-f'' + V f = \lambda f\) with \(\varphi_\lambda (0) = \psi_\lambda'(0) = 1\), \(\varphi_\lambda' (0) = \psi_\lambda(0) = 0\). In a semibounded (or even non-negative) setting, a different approach is given by the sesquilinear forms \({\mathfrak t}[f,g] := \int_0^\infty (f'\bar{g}' + V f\bar{g}) \, dx\) and \({\mathfrak t}_\tau[f,g] := {\mathfrak t}[f,g] + \tau f(0)\overline{g(0)}\) on suitable domains. This form approach leads to the study of the Friedrichs and the Krein-von Neumann extension of \(S\). Here, the so-called boundary pair \(({\mathbb C}, \, \Lambda)\) with the mapping \(\Lambda: \operatorname{dom} {\mathfrak t} \rightarrow {\mathbb C}\), \(\Lambda f := f(0)\), establishes a connection to the above boundary triplet. Now, the abstract theory of the monograph can be regarded more or less as a generalization of these considerations (and a lot more) to the setting of an arbitrary symmetric operator \(S\) (or, even more generally, relation \(S\)) with equal defect numbers in a Hilbert space \({\mathfrak H}\).

The first part of the monograph consists of Chapters 1–5, presenting the abstract theory. Chapter 1 is in some sense is preparatory, but it also contains a lot of material on linear operators and relations which goes beyond the “standard knowledge”, e.g., decomposition of linear relations, graph and resolvent convergence of operators and relations, parametric representations, and Nevanlinna families. All this material is used later in the text. Chapter 2 goes to the heart of the matter, here boundary triplets, Weyl functions and their properties are studied. An essential point in this chapter is Krein’s formula in a very general context. Chapter 3 continues the theme of Chapter 2; here, the analytic properties of the Weyl function are related to the resolvent and the spectrum of a self-adjoint extension. Chapter 4 is about a closely related inverse problem: a given Nevanlinna function is interpreted as a Weyl function of a boundary triplet. Corresponding models are often constructed with the help of reproducing kernel Hilbert spaces, a topic that is also discussed in Chapter 4. Finally, Chapter 5 complements the earlier chapters by discussing semibounded extensions of semibounded operators or relations in detail. Here, the key feature is the notion of a boundary pair, which connects boundary triplet and form techniques. Much of this material seems to be new in the mathematical literature.

The second part of the monograph is of a more applied nature. In Chapters 6–8, the authors show how the abstract methods fit with concrete boundary value problems. The classical and standard example of a Sturm-Liouville operator (as indicated above) is discussed in Chapter 6 in full detail. Of particular interest is the interplay with boundary pairs, boundary triplets and form techniques in this chapter. The next chapter treats a class of first order systems of differential equations. Here, it becomes clear that multivalued operators, i.e., linear relations, arise naturally in applications. Finally, Chapter 8 studies Schrödinger operators on bounded domains. A complete systematic description of all proper boundary conditions that lead to self-adjoint (or maximal dissipative/accumulative) realizations of an elliptic PDE (and their spectral analysis) is an interesting topic. The boundary triplet method is obviously most suitable and shows its full power here. The monograph also contains some Appendices to keep the text self-contained, an impressive List of References, and some Notes and interesting Historical Comments.

Reviewer: Andreas Fleige (Dortmund)

### MSC:

47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |

47Axx | General theory of linear operators |

47Bxx | Special classes of linear operators |

47E05 | General theory of ordinary differential operators |

47Fxx | Partial differential operators |

34Bxx | Boundary value problems for ordinary differential equations |

34Lxx | Ordinary differential operators |

35Pxx | Spectral theory and eigenvalue problems for partial differential equations |

93Bxx | Controllability, observability, and system structure |