This book contains extended lecture notes for an advanced graduate course taught at Vienna University. The main subjects are wave radiation and spectral and scattering theories for the Schrödinger and Klein-Gordon equations: \[ \begin{aligned} i\dot{\psi} (x,t)&=H\psi(x,t):=-\Delta\psi (x,t)+V(x)\psi(x,t),\\ \ddot{\psi}(x,t)&=\Delta\psi(x,t)-m^2\psi(x,t)-V(x)\psi(x,t)\end{aligned} \] for \(x\in\mathbb R^3\), where \(V\) is real-valued, continuous and \(V(x)\langle x\rangle^\beta\) is bounded for some \(\beta>0\). (The authors define \(\langle x\rangle=(1+| x|)^{1/2}\) but possibly they intended \(\langle x\rangle =(1+| x|^2)^{1/2}\).)
The book consists of 42 sections grouped into 11 chapters. The first two sections give basic concepts (such as Fourier transforms of distributions and some aspects of pseudodifferential operators) with references. Fundamental properties of the Schrödinger equation are derived in Sections 3–15. Section 15 establishes a result of T. Kato which can be stated roughly as follows: If \(\beta>1\) then \(H\) has no positive eigenvalues. (References are given in the text.) In Sections 16–22, the Agmon-Jensen-Kato spectral theory of dispersion decay in weighted Sobolev norms is presented for the first time in a textbook. Theorem 17.1, due to A. Jensen and Kato, concerns the asymptotic decay of the resolvent operator \(R(\omega)=(H-\omega)^{-1}\) as \(|\omega|\to\infty\) when \(R(\omega)\) is regarded as an operator on certain weighted Sobolev spaces. In Section 19, a “limiting absorption principle” is established: it states that the limits \(R(\lambda\pm i\epsilon)\to R(\lambda\pm 0)\) as \(\epsilon\to 0=^+\) exist in an appropriate operator norm for \(\lambda\geq 0\), provided \(\beta>2\) and 0 is neither an eigenvalue nor a resonance for H. Theorem 17.1 and the limiting absorption principle are relevant to dispersion decay since
\[ \psi(t)=(2\pi i)^{-1}\int_0^\infty e^{-i\omega t}[R(\omega +i0)-R(\omega -0)]\psi (0)d\omega \] if \(\psi(0)\) is in the subspace \(X_c\) of continuity of \(H\). The main dispersion decay result, Theorem 21.1, is due to Jensen and Kato; its statement is as follows: If \(\beta>3\) and 0 is neither an eigenvalue nor a resonance of \(H\) and if \(\psi (0)\in X_c\) then
\[ \|\langle x\rangle^{-\sigma}\psi(x,t)\|\leq C(\sigma)\langle t\rangle^{-3/2}\|\langle x\rangle^\sigma \psi(x,0)\|,\quad \sigma >5/2 \]
where \(\|\cdot\|\) denotes the \(L^2(\mathbb R^3,dx)\) norm. The proof relies not only on the asymptotic decay of \(R(\omega)\) as \(|\omega|\to\infty\) but also on the asymptotics as \(\omega\to\infty\).
In Sections 23–33 the dispersion decay is applied to scattering theory and in particular to the completeness of the wave operators, to the diagonalization of the Schrödinger and scattering operators, and to a novel discussion of the scattering cross section. The final two chapters, 10 and 11, deal with recent results of the authors which involve extending the Agmon-Jensen-Kato approach to the Klein-Gordon equation (Chapter 10) and to the wave equation (Chapter 11). This extension is not routine, because the high-energy asymptotics for the resolvent operator are different from those for the Schrödinger equation. The book ends with an appendix on the Sobolev Embedding Theorem, in Section 42.
The book is carefully written, features “complete and streamlined proofs”, and some material, such as a novel justification of the “limiting amplitude principle”, appears here for the first time.