This chapter addresses generalizations of the Schrödinger equation. It tries to convey that the Schrödinger equation is not the whole story when it comes to quantum physics. This is illustrated by expanding the framework in two rather orthogonal directions: relativistic quantum physics and open quantum systems. The former is introduced by taking the Klein–Gordon equation as the starting point, before shifting attention to the Dirac equation. Its time-independent version is solved numerically for a one-dimensional example, and its relation to the Schrödinger equation is derived. Also here, the Pauli matrices play crucial roles. The notion of open quantum systems is motivated by the fact that it is hard to keep a quantum system completely isolated from its surroundings – and that this necessitates a different approach than the one provided by wave functions. To this end, reduced density matrices and the notion of master equations are introduced. It is explained why master equations of the form of the generic Gorini–Kossakowski–Lindblad–Sudarshan (GKLS) equation are desirable. Two particular phenomena following this equation are studied quantitatively: amplitude damping for a single quantum bit system and particle capture in a confining potential. Again, these examples draw directly on previous ones.

We explain in detail the strategy of Buslaev--Perelman--Sulem (BPS): symplectic projection of a trajectory on a solitary manifold andmodulation equations for the projection and time decay for a transversal component using the Poincar\'e normal form and Fermi Golden Rule for the transversal dynamics. We present an extensive list of results onasymptotic stability of stationary orbitsand solitons that rely on the BPS strategy and its generalizations byS. Cuccagna, Y. Martel, F. Merle, T. Mizumachi, K. Nakanishi, I. Rodnianski,W. Schlag,I. M. Sigal, A. Soffer,R. L. Pego,T. P. Tsai,M. I. Weinstein, H. T. Yau, and others. We also mention the results on stability and instability ofself-similar, spherically symmetric solutions and rotating Kerr solutions ofequations of the General Theory of Relativity by T. Harada, C. E. Kenig, H. Maeda,F. Merle, W. Schlag, and others. Moreover, we illustrate the BPS strategyin the simplest modelof a 1D Schrödinger equation coupled to a nonlinear oscillator, giving complete proofs with all details.