Semilinear Schrödinger equations.

*(English)*Zbl 1055.35003
Courant Lecture Notes in Mathematics 10. Providence, RI: American Mathematical Society (AMS); New York, NY: Courant Institute of Mathematical Sciences (ISBN 0-8218-3399-5/pbk). xiii, 323 p. (2003).

This book deals with the semilinear Schrödinger equation and its associated Cauchy problem
\[
iu_t+\Delta u+g(u)=0,\qquad u(0)=\varphi.
\]
From the applications point of view, such a problem is important for instance in nonlinear optics and quantum field theory. From the mathematical point of view it is an intriguing and challenging problem, which possesses a mixture of the properties of parabolic and hyperbolic equations. It is also linked to other dispersive equations such as the Korteweg-de Vries equation.

The book is based on the notes of several courses given by the author between 1989 and 1997. It touches all the major aspects and topics of the theory, such as local existence, uniqueness, well-posedness, regularity, global existence, and asymptotic behavior. In particular, emphasis is put on energy and Strichartz’s estimates which turn out to be fundamental tools for the analysis. The book, written by one of the leading expert on the subject, is also an up-to-date source of references for recent results and open problems, well represented in its extensive bibliography. It can certainly be used as a guideline for a course to an audience with a sufficient background on functional analysis and partial differential equations.

Here are some details on the chapters.

Chapter 1. Preliminary results. This introductory chapter reports some basic notions and results, which will be useful in the sequel. In particular, they concern functional analysis, Sobolev spaces, elliptic equations, and semigroups of linear operators.

Chapter 2. The linear Schrödinger equation. The linear equation is studied in the case \(\Omega=\mathbb R^N\), stating and proving some fundamental properties, as, first of all, Strichartz’s estimates on the solutions. It is then showed how, from Strichartz’s estimates, some regularizing effects and some properties of spatial decay follow.

Chapter 3. The Cauchy problem in a general domain. The nonlinear equation on a general domain \(\Omega\subset\mathbb R^N\) is studied. Some examples of typical nonlinearities are given, in particular the pure power nonlinearity \(g(u)=\lambda| u| ^\alpha u\) where \(\lambda\in\mathbb R\), and \(\alpha\geq0\). In the case under consideration some problems arise, due to the nonlinearity and to the fact that, if \(\Omega\neq\mathbb R^N\), then the Strichartz’s estimates do not in general hold. Some suitable conservation laws for the solutions (norm in \(L^2\), and some types of energy in \(H^1_0\)) suggest to use \(L^2\) or \(H^1_0\) as “energy space” for studying the local existence of the solutions. It is shown how the energy estimates technique permits to obtain the local existence of solutions and, under the additional hypothesis of uniqueness, the well-posedness of the problem. The additional hypothesis of uniqueness in order to get the well-posedness, is justified by the fact that the known uniqueness results are obtained by techniques that strongly depend on the nonlinearity and on the geometry of the domain \(\Omega\).

Chapter 4. The local Cauchy problem. The nonlinear Cauchy problem, in the case \(\Omega=\mathbb R^N\) is studied. In this case Strichartz’s estimates hold, and hence uniqueness of the local solution is proved. This result holds for a rather large class of nonlinearities. However, for some particular cases, such results do not hold. Hence a technique due to Kato, based on a fixed point argument and Strichartz’s estimates, is used to prove local existence and uniqueness, as well as well-posedness. The existence and properties of solutions in \(H^s(\mathbb R^N)\) with \(s<N/2\) (using estimates in a suitable Besov space), in \(H^m(\mathbb R^n)\) with \(m<N/2\) and integer, as well as for nonautonomous Schrödinger equations, are also investigated.

Chapter 5. Regularity and the smoothing effect. The problem of having more regularity for the solution as the initial datum is more regular is studied. Results are given for the regularity in several \(H^s\) spaces, for various values of \(s\). Finally, the smoothing effect is studied.

Chapter 6. Global existence and finite time blow-up. Results are given for the global existence and for the global non existence (blow-up) of the solutions. The existence results are given for nonlinearities for which the conservation of energy is assured, and for initial data having suitable properties (such as smallness, oscillation, and asymptotic homogeneity). The nonexistence results are given for suitable nonlinearities and initial data too.

Chapter 7. Asymptotic behavior in the repulsive case. This chapter is based on the scattering theory. For a given initial datum \(\varphi\) its scattering states are suitable asymptotic limits (as \(t\to\pm\infty)\)) of the corresponding global solution. Results are given for existence, uniqueness and non existence of the scattering states in some functional spaces such as \(L^2\), \(H^1\) and a suitable weighted intersection of them. Moreover, using the Morawetz’s estimate, the decay of the solution is proved for a suitable case. Chapter 8. Stability of bound states in the attractive case. In this chapter, concerning the case of pure power nonlinearity with \(\lambda>0\), and a non vanishing initial datum \(\varphi\in H^1\), the existence of solution of the form \(e^{i\omega t}\varphi(x)\) (stationary states) is investigated. Moreover, results for instability as well as stability (in some suitable senses) of the stationary states are given.

Chapter 9. Further results. In this chapter, some further extensions to other (physically relevant) situations are given, such as presence of magnetic field, quadratic potential and logarithmic Schrödinger equation.

Finally, all the chapters end with a section of comments, usually showing extensions and generalizations of the results to other situations.

The book is based on the notes of several courses given by the author between 1989 and 1997. It touches all the major aspects and topics of the theory, such as local existence, uniqueness, well-posedness, regularity, global existence, and asymptotic behavior. In particular, emphasis is put on energy and Strichartz’s estimates which turn out to be fundamental tools for the analysis. The book, written by one of the leading expert on the subject, is also an up-to-date source of references for recent results and open problems, well represented in its extensive bibliography. It can certainly be used as a guideline for a course to an audience with a sufficient background on functional analysis and partial differential equations.

Here are some details on the chapters.

Chapter 1. Preliminary results. This introductory chapter reports some basic notions and results, which will be useful in the sequel. In particular, they concern functional analysis, Sobolev spaces, elliptic equations, and semigroups of linear operators.

Chapter 2. The linear Schrödinger equation. The linear equation is studied in the case \(\Omega=\mathbb R^N\), stating and proving some fundamental properties, as, first of all, Strichartz’s estimates on the solutions. It is then showed how, from Strichartz’s estimates, some regularizing effects and some properties of spatial decay follow.

Chapter 3. The Cauchy problem in a general domain. The nonlinear equation on a general domain \(\Omega\subset\mathbb R^N\) is studied. Some examples of typical nonlinearities are given, in particular the pure power nonlinearity \(g(u)=\lambda| u| ^\alpha u\) where \(\lambda\in\mathbb R\), and \(\alpha\geq0\). In the case under consideration some problems arise, due to the nonlinearity and to the fact that, if \(\Omega\neq\mathbb R^N\), then the Strichartz’s estimates do not in general hold. Some suitable conservation laws for the solutions (norm in \(L^2\), and some types of energy in \(H^1_0\)) suggest to use \(L^2\) or \(H^1_0\) as “energy space” for studying the local existence of the solutions. It is shown how the energy estimates technique permits to obtain the local existence of solutions and, under the additional hypothesis of uniqueness, the well-posedness of the problem. The additional hypothesis of uniqueness in order to get the well-posedness, is justified by the fact that the known uniqueness results are obtained by techniques that strongly depend on the nonlinearity and on the geometry of the domain \(\Omega\).

Chapter 4. The local Cauchy problem. The nonlinear Cauchy problem, in the case \(\Omega=\mathbb R^N\) is studied. In this case Strichartz’s estimates hold, and hence uniqueness of the local solution is proved. This result holds for a rather large class of nonlinearities. However, for some particular cases, such results do not hold. Hence a technique due to Kato, based on a fixed point argument and Strichartz’s estimates, is used to prove local existence and uniqueness, as well as well-posedness. The existence and properties of solutions in \(H^s(\mathbb R^N)\) with \(s<N/2\) (using estimates in a suitable Besov space), in \(H^m(\mathbb R^n)\) with \(m<N/2\) and integer, as well as for nonautonomous Schrödinger equations, are also investigated.

Chapter 5. Regularity and the smoothing effect. The problem of having more regularity for the solution as the initial datum is more regular is studied. Results are given for the regularity in several \(H^s\) spaces, for various values of \(s\). Finally, the smoothing effect is studied.

Chapter 6. Global existence and finite time blow-up. Results are given for the global existence and for the global non existence (blow-up) of the solutions. The existence results are given for nonlinearities for which the conservation of energy is assured, and for initial data having suitable properties (such as smallness, oscillation, and asymptotic homogeneity). The nonexistence results are given for suitable nonlinearities and initial data too.

Chapter 7. Asymptotic behavior in the repulsive case. This chapter is based on the scattering theory. For a given initial datum \(\varphi\) its scattering states are suitable asymptotic limits (as \(t\to\pm\infty)\)) of the corresponding global solution. Results are given for existence, uniqueness and non existence of the scattering states in some functional spaces such as \(L^2\), \(H^1\) and a suitable weighted intersection of them. Moreover, using the Morawetz’s estimate, the decay of the solution is proved for a suitable case. Chapter 8. Stability of bound states in the attractive case. In this chapter, concerning the case of pure power nonlinearity with \(\lambda>0\), and a non vanishing initial datum \(\varphi\in H^1\), the existence of solution of the form \(e^{i\omega t}\varphi(x)\) (stationary states) is investigated. Moreover, results for instability as well as stability (in some suitable senses) of the stationary states are given.

Chapter 9. Further results. In this chapter, some further extensions to other (physically relevant) situations are given, such as presence of magnetic field, quadratic potential and logarithmic Schrödinger equation.

Finally, all the chapters end with a section of comments, usually showing extensions and generalizations of the results to other situations.

Reviewer: Fabio Bagagiolo (Trento)

##### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35Q55 | NLS equations (nonlinear Schrödinger equations) |

35A05 | General existence and uniqueness theorems (PDE) (MSC2000) |

35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |

35B35 | Stability in context of PDEs |

35B40 | Asymptotic behavior of solutions to PDEs |

35B65 | Smoothness and regularity of solutions to PDEs |

35P25 | Scattering theory for PDEs |

35J10 | Schrödinger operator, Schrödinger equation |

35Q40 | PDEs in connection with quantum mechanics |

78A60 | Lasers, masers, optical bistability, nonlinear optics |

81V80 | Quantum optics |