##
**Periodic differential operators.**
*(English)*
Zbl 1267.34001

Operator Theory: Advances and Applications 230. Basel: Birkhäuser (ISBN 978-3-0348-0527-8/hbk; 978-3-0348-0528-5/ebook). viii, 216 p. (2013).

The book, which is a comprehensive presentation of the theory of periodic differential operators, is motivated from the fact that during the past decades there has been an increasing progress in the study of such operators. And, this study is leading to new developments, as well as to some changes in perspective, as, for example, it happened with the Schrödinger and the Dirac operators. Although some of the spectral properties of these two operators are very different, they have much in common when expressed in the form of a Hamiltonian system. Also, the use of oscillation properties of solutions of differential equations has proved to be an effective tool in the study of the spectral properties of the associated differential operators, beginning from Sturm’s and Prüfer’s results.

The book lays out the theoretical foundations and then moves on to give an account of more recent results, relating, in particular, to the eigenvalues and the spectral theory of the Hill and Dirac equations. It consists of five chapters, and all chapters close with a great number of additional explanatory notes connecting the results with a thorough bibliography.

Chapter 1 presents a brief summary of the necessary concepts and results from the theory of ordinary differential equations, in order to introduce the terminology for establishing (in section 1.3) the existence of Floquet solutions of periodic systems of the form \[ u'=Au,\tag{s} \] where \(A\) is a complex matrix valued periodic function. The \(2\times2\)-dimensional case of \(A\), when it is written in the form \(A=JS\), with \(J=\left(\begin{smallmatrix} 0&1\\-1&0\end{smallmatrix}\right)\), is discussed in section 1.4, where the role of the Hill discriminant to the existence of periodic and semi-periodic solutions is emphasized. The case of Hill’s equation and periodic Dirac systems, which include a spectral parameter, are studied in sections 1.5, 1.6 (and in section 2.8 with even coefficients) for the equation \[ u'=J(B+\lambda W)u.\tag{\(*\)} \] The Fourier method is used in section 1.7 to show that the Mathieu equation \[ y''(x)+(\lambda-2\cos2x)y(x)=0,\tag{M} \] has no points of coexistence for any value of the parameter \(c \neq 0\). Spectral properties of equation \((*)\), associated with a boundary condition of the form \(u(a)=\omega u(0)\), are investigated in section 1.8. In an appendix to section 1.9, the so-called Rofe-Beketov formula for the solutions of systems like \((s)\) for the case \(\operatorname{Tr}(a)=0\) is studied.

Chapter 2 starts with section 2.2, where a discussion of the Prüfer equations for the \(2\times2\) system \((*)\) is presented. From section 2.3 onwards, the assumption that the system is either a Sturm-Liouville, or a Dirac system is often made. In Section 2.3, it is emphasized that the Prüfer angle depends monotonically on the spectral parameter \(\lambda\). The section closes with a relative oscillation theorem, where a method of counting eigenvalues in an interval with separated boundary conditions, is shown. The rotation number, which is the growth-rate of the oscillations of the solutions to periodic systems, is a continuous, monotonously increasing function of the real spectral parameter, and its properties are discussed in Section 2.4. This is used in Section 2.5 to obtain information about the number of zeros of the eigenfunctions of BVPs under periodic, or semi-periodic conditions. In Section 2.6, the upper endpoint of the \(n\)-th stability interval is discussed, while in Section 2.7 an example with a step-function is given. Some useful results implied by the comparison of eigenvalues are given in Section 2.9, and the chapter closes with some estimates on the least eigenvalue for problems with boundary conditions as in Section 1.8.

The main purpose of Chapter 3 is to examine the nature of the instability intervals – first, their asymptotic lengths as they recede to infinity and, second, the more specialized situation, when there is only a finite number of them are present. To deal with the lengths, the authors require asymptotic estimates of the eigenvalues. The theory of the first two chapters provides two methods to produce these estimates. The first method used in this chapter is based on the Prüfer transformation and oscillation theory of Chapter 2. The other method uses a direct examination of the discriminant, as it is described in Chapter 1. In Section 3.2, a Prüfer-type transformation is given for the equation \[ y''+(\lambda w-q)y=0,\tag{\(**\)} \] which is then applied in Section 3.3. There are no assumptions on the differentiability of \(w\) or \(q\). The same transformation is applied in Section 3.4 to obtain Titchmarsh’s asymptotic formula for the eigenvalues of \((**)\), while similar results are given in Section 3.5 under some differentiability conditions on \(q\). Section 3.6 is devoted to the length \(l_n\) of the instability intervals. Indeed, the authors prove the following result.

(Theorem 3.6.1) Let \(w=1\), and let \(q\) be infinitely differentiable with \(|q^{(r)}(x)|\leq Kr!d^{-r}\) for all \(x\) and \(r\geq 0\), where \(K\) and \(d\) are independent of \(x\) and \(r\). Then there are positive constants \(A\) and \(B\) such that \(l_n\leq A\exp(-Bn)\) as \(n\to +\infty\).

A similar result concerning the length \(l(\mu)\) of an instability interval with mid-point \(\mu\) is, also, given. A precise asymptotic expression for the length of the instability interval in the case of the Mathieu equation \((M)\) is given in Section 3.7, while in Section 3.8, asymptotic formulae for the basic solutions of \((**)\) with \(w=1\) are obtained. Sections 3.9–3.12 cover the problem of determining the special nature of \(q\) for equation \((**)\), given that only a finite number of the instability intervals are nonempty.

Chapter 4 begins with the observation that the eigenvalues and eigenfunctions of these boundary-value problems give rise to a generalization of Fourier series convergent in the sense of a suitable Hilbert space. Then, the study proceeds to the spectral properties of periodic equations on the real line and on a half-line with a boundary condition of separated type. This is done in Section 4.3, where the limiting process of sending one of the endpoints of a bounded interval to infinity is studied, finding a limit for the spectral functions governing the generalized Fourier transforms of the boundary-value problems on the interval. This limit is a spectral function for the half-line problem. The associated self-adjoint operator, constructed in Section 4.4, will in general have intervals of continuous spectrum in addition to eigenvalues. In the special case of periodic equations on the half-line, it is shown in Section 4.5 that the spectrum consists of purely absolutely continuous bands coinciding with the closure of the stability set, with at most one eigenvalue in each instability interval. The chapter closes with an extension of the oscillation method for counting eigenvalues, introduced in Section 2.3 for periodic separated boundary-value problems, to the half-line case.

Chapter 5 begins by remarking in Section 5.2 that the spectral bands remain intervals of purely absolutely continuous spectrum under a very mild decay condition on the perturbation. In Section 5.3, it is observed that, if the perturbation tends to zero at infinity, then every compact subinterval of an instability interval contains at most finitely many eigenvalues and no further spectrum. In particular, this means that the instability intervals, while not devoid of spectrum in general, continue to be gaps in the essential spectrum. Also, the asymptotics for the distribution of the eigenvalues thus introduced into the gaps in the limit of slow variation of the perturbation are derived. The question of whether an instability interval, as a whole, contains a finite, or infinite number of eigenvalues, turns out to have a more subtle answer, given in Section 5.4. There is a critical boundary case for perturbations with \(x^{-2}\) asymptotic decay, and the critical coupling constant can be expressed in terms of the derivative of Hill’s discriminant at the point of transition between instability and stability. In the supercritical case, where the eigenvalues in the gap accumulate at a band edge, their asymptotic distribution is obtained in Section 5.5, which shows that they are exponentially close to the band.

The book is well written and will be valuable to students and academics both for general reference and as an introduction to topics related to active research. It contains a rich list of 210 references of carefully selected old and new works, which, also, might be a good source to graduates for further study of the theory of periodic differential operators.

The book lays out the theoretical foundations and then moves on to give an account of more recent results, relating, in particular, to the eigenvalues and the spectral theory of the Hill and Dirac equations. It consists of five chapters, and all chapters close with a great number of additional explanatory notes connecting the results with a thorough bibliography.

Chapter 1 presents a brief summary of the necessary concepts and results from the theory of ordinary differential equations, in order to introduce the terminology for establishing (in section 1.3) the existence of Floquet solutions of periodic systems of the form \[ u'=Au,\tag{s} \] where \(A\) is a complex matrix valued periodic function. The \(2\times2\)-dimensional case of \(A\), when it is written in the form \(A=JS\), with \(J=\left(\begin{smallmatrix} 0&1\\-1&0\end{smallmatrix}\right)\), is discussed in section 1.4, where the role of the Hill discriminant to the existence of periodic and semi-periodic solutions is emphasized. The case of Hill’s equation and periodic Dirac systems, which include a spectral parameter, are studied in sections 1.5, 1.6 (and in section 2.8 with even coefficients) for the equation \[ u'=J(B+\lambda W)u.\tag{\(*\)} \] The Fourier method is used in section 1.7 to show that the Mathieu equation \[ y''(x)+(\lambda-2\cos2x)y(x)=0,\tag{M} \] has no points of coexistence for any value of the parameter \(c \neq 0\). Spectral properties of equation \((*)\), associated with a boundary condition of the form \(u(a)=\omega u(0)\), are investigated in section 1.8. In an appendix to section 1.9, the so-called Rofe-Beketov formula for the solutions of systems like \((s)\) for the case \(\operatorname{Tr}(a)=0\) is studied.

Chapter 2 starts with section 2.2, where a discussion of the Prüfer equations for the \(2\times2\) system \((*)\) is presented. From section 2.3 onwards, the assumption that the system is either a Sturm-Liouville, or a Dirac system is often made. In Section 2.3, it is emphasized that the Prüfer angle depends monotonically on the spectral parameter \(\lambda\). The section closes with a relative oscillation theorem, where a method of counting eigenvalues in an interval with separated boundary conditions, is shown. The rotation number, which is the growth-rate of the oscillations of the solutions to periodic systems, is a continuous, monotonously increasing function of the real spectral parameter, and its properties are discussed in Section 2.4. This is used in Section 2.5 to obtain information about the number of zeros of the eigenfunctions of BVPs under periodic, or semi-periodic conditions. In Section 2.6, the upper endpoint of the \(n\)-th stability interval is discussed, while in Section 2.7 an example with a step-function is given. Some useful results implied by the comparison of eigenvalues are given in Section 2.9, and the chapter closes with some estimates on the least eigenvalue for problems with boundary conditions as in Section 1.8.

The main purpose of Chapter 3 is to examine the nature of the instability intervals – first, their asymptotic lengths as they recede to infinity and, second, the more specialized situation, when there is only a finite number of them are present. To deal with the lengths, the authors require asymptotic estimates of the eigenvalues. The theory of the first two chapters provides two methods to produce these estimates. The first method used in this chapter is based on the Prüfer transformation and oscillation theory of Chapter 2. The other method uses a direct examination of the discriminant, as it is described in Chapter 1. In Section 3.2, a Prüfer-type transformation is given for the equation \[ y''+(\lambda w-q)y=0,\tag{\(**\)} \] which is then applied in Section 3.3. There are no assumptions on the differentiability of \(w\) or \(q\). The same transformation is applied in Section 3.4 to obtain Titchmarsh’s asymptotic formula for the eigenvalues of \((**)\), while similar results are given in Section 3.5 under some differentiability conditions on \(q\). Section 3.6 is devoted to the length \(l_n\) of the instability intervals. Indeed, the authors prove the following result.

(Theorem 3.6.1) Let \(w=1\), and let \(q\) be infinitely differentiable with \(|q^{(r)}(x)|\leq Kr!d^{-r}\) for all \(x\) and \(r\geq 0\), where \(K\) and \(d\) are independent of \(x\) and \(r\). Then there are positive constants \(A\) and \(B\) such that \(l_n\leq A\exp(-Bn)\) as \(n\to +\infty\).

A similar result concerning the length \(l(\mu)\) of an instability interval with mid-point \(\mu\) is, also, given. A precise asymptotic expression for the length of the instability interval in the case of the Mathieu equation \((M)\) is given in Section 3.7, while in Section 3.8, asymptotic formulae for the basic solutions of \((**)\) with \(w=1\) are obtained. Sections 3.9–3.12 cover the problem of determining the special nature of \(q\) for equation \((**)\), given that only a finite number of the instability intervals are nonempty.

Chapter 4 begins with the observation that the eigenvalues and eigenfunctions of these boundary-value problems give rise to a generalization of Fourier series convergent in the sense of a suitable Hilbert space. Then, the study proceeds to the spectral properties of periodic equations on the real line and on a half-line with a boundary condition of separated type. This is done in Section 4.3, where the limiting process of sending one of the endpoints of a bounded interval to infinity is studied, finding a limit for the spectral functions governing the generalized Fourier transforms of the boundary-value problems on the interval. This limit is a spectral function for the half-line problem. The associated self-adjoint operator, constructed in Section 4.4, will in general have intervals of continuous spectrum in addition to eigenvalues. In the special case of periodic equations on the half-line, it is shown in Section 4.5 that the spectrum consists of purely absolutely continuous bands coinciding with the closure of the stability set, with at most one eigenvalue in each instability interval. The chapter closes with an extension of the oscillation method for counting eigenvalues, introduced in Section 2.3 for periodic separated boundary-value problems, to the half-line case.

Chapter 5 begins by remarking in Section 5.2 that the spectral bands remain intervals of purely absolutely continuous spectrum under a very mild decay condition on the perturbation. In Section 5.3, it is observed that, if the perturbation tends to zero at infinity, then every compact subinterval of an instability interval contains at most finitely many eigenvalues and no further spectrum. In particular, this means that the instability intervals, while not devoid of spectrum in general, continue to be gaps in the essential spectrum. Also, the asymptotics for the distribution of the eigenvalues thus introduced into the gaps in the limit of slow variation of the perturbation are derived. The question of whether an instability interval, as a whole, contains a finite, or infinite number of eigenvalues, turns out to have a more subtle answer, given in Section 5.4. There is a critical boundary case for perturbations with \(x^{-2}\) asymptotic decay, and the critical coupling constant can be expressed in terms of the derivative of Hill’s discriminant at the point of transition between instability and stability. In the supercritical case, where the eigenvalues in the gap accumulate at a band edge, their asymptotic distribution is obtained in Section 5.5, which shows that they are exponentially close to the band.

The book is well written and will be valuable to students and academics both for general reference and as an introduction to topics related to active research. It contains a rich list of 210 references of carefully selected old and new works, which, also, might be a good source to graduates for further study of the theory of periodic differential operators.

Reviewer: George Karakostas (Ioannina)

### MSC:

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

34L15 | Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators |

34L10 | Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators |

34C20 | Transformation and reduction of ordinary differential equations and systems, normal forms |

34A30 | Linear ordinary differential equations and systems |

34B30 | Special ordinary differential equations (Mathieu, Hill, Bessel, etc.) |

34L05 | General spectral theory of ordinary differential operators |

34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |

47A55 | Perturbation theory of linear operators |

47E05 | General theory of ordinary differential operators |

81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |

81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |

34L16 | Numerical approximation of eigenvalues and of other parts of the spectrum of ordinary differential operators |