The Schrödinger and Riccati equations
The major topics of the book are: (i) Linear Schrödinger equation, (ii) The nonlinear Schrödinger equation, (iii) The Riccati equation and (iv) The Schrödinger-Riccati equation. The authors introduce the Schrödinger equation as the reduction of the generalized Dirac-Breit equation for a many-particle system in the nonrelativistic form, (though the title of the chapter is ``Derivation of the Schrödinger equation, in the text it is mentioned that the equation cannot be derived). In the next chapter, local and global properties, in particular the viral theorem, are discussed in the configuration space. This is followed by observations on the momentum representation of both the equation and the wave function by Fourier transformation. This is illustrated with some representative numerical calculations. In the chapter: The local Schrödinger equation, the topics discussed are alternatives for the computational solution of the equation (local energy methods) and a survey of alternative numerical methods for global solution. The following chapter, The time-dependent Schrödinger equation, presents the three representations namely (a) Schrödinger, (b) Heisenberg and (c) interaction representation (Dirac), time evolution in interaction representation and transition probabilities are also given. In the brief part, The nonlinear Schrödinger equation, the discussion is confined only on conservation laws and conserved quantities, and numerical solution to the (one-dimensional) equation. The only parts of the book concomitant to the title of the book are the last parts. The brief penultimate part, The Riccati equation, consists of the well-known reduction of the linear homogeneous second-order equation to the Riccati equation, followed by a discussion on some of the properties of its solution. The last part namely, the Schrödinger-Riccati equation is an essentially quantum mechanical application of the Riccati equation. This deals with an alternative to the perturbation theory which may be more apt to application in many-body systems. It consists of starting intuitively a tolerably good approximate wave function of the exact many-body wave function and then obtain a Riccati equation (quadratic term only) for the difference of the exact and the intuitive approximate wave function. Finally, express the equation in a power of this difference (mentioned as the Schrödinger-Riccati equation). The last chapter attempts to complement and illustrate with practical detail of the formulation and theoretical development with numerical calculations. The presentation followed is quite formal, in particular, the last part, the iteration and series expansions are justified only with the `pious' hope that it will lead to correct and exact results (convergence?). There are quite a few incomplete and incoherent statements. The text is full of references almost in every line, which all added is a considerable portion of the total volume (218 p.) of the text. This is apart from 31 pages of references and the bibliography at the end. It is not clear to whom the book is addressed.
scientific article; zbMATH DE number 1419150