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Zbl 1130.81001
Dong, Shi-Hai
Factorization method in quantum mechanics. (English)
[B] Fundamental Theories of Physics 150. Dordrecht: Springer. xix, 297~p. EUR~119.95/net; SFR~197.00; \sterling~92.50; \$~169.00 (2007). ISBN 978-1-4020-5795-3/hbk

In conceiving this book the author realizes that, in spite of its general relevance, there are relatively few tutorial resources dealing with the application of the factorization method to Quantum Mechanical problems. His aim is, therefore, to provide an up-to-date organized account of material that can be addressed to an interdisciplinary graduate-level audience. In doing so, he discusses a number of quantum-mechanical problems of interest in physics and chemistry by paying prior attention to the identification of the first-order ladder operators yielding the Hamiltonian operators into a factorized form, and to the characterization of the dynamical algebras they may form together with other operators. Besides the elegance and the effectiveness in characterizing matrix-elements, a motivation for the algebraic approach also relies in the pedagogical expectation that beginners can be better driven to topics like coherent states and supersymmetric Quantum Mechanics. The book is divided into six Parts: I. Introduction, II. Method, III. Applications in Non-relativistic Quantum Mechanics, IV. Applications in Relativistic Quantum Mechanics, V. Quantum Control, VI. Conclusions and outlooks. In Part I, the author argues on historical development of the subject. Part II simply consists in a couple of pages introducing to the idea underlying the method, and in a chapter giving fundamentals on groups and algebras. Parts III and IV define the actual core of the monograph since they are concerned with the study of explicit problems, and the discussions of the corresponding dynamical algebras of the su(2) and su(1,1) type. Parts III and IV consist in several chapters, each of them considering in some detail a given second-order ordinary differential operator. Chapter headings for parts III and IV are: 4. The harmonic oscillator, 5. The infinitely deep square-well potential, 6. Morse potential, 7. Pösch potential, 8. Pseudoharmonic potential, 9. Algebraic approach to an electron in a uniform magnetic field, 10. Ring-shaped non-spherical oscillator, 11. Generalized Laguerre functions, 12. New noncentral ring-shaped potential, 13. Pösch-Teller like potential, 14. Position-dependent mass Schrödinger equation for a singular oscillator, 15. SUSYQM and SWKB approach to the Dirac equation with a Coulomb potential in 2+1 dimensions, 16. Realization of dynamic group for the Dirac Hydrogen-Like atom in 2+1 dimension, 17. Algebraic approach to Klein-Gordon equation with the hydrogen-like atom in 2+1 dimensions, 18. SUSYQM and SWKB approaches to Dirac and Klein-Gordon equations with hyperbolic potential. As for Part V, the author argues on the controllability of quantum systems in the presence of potentials of the Morse and the Pösch-Teller types. Conclusions and outlooks are given in two pages defining Part VI (Chapter 21). Five Appendices (A. Integral formulas of the confluent hypergeometric functions, B. Mean values $\overline{r^{k}}$ for hydrogen-like atom, C. Commutator identities, D. Angular momentum operators in spherical coordinates, E. Confluent hypergeometric function) and a references list of 698 items, that gives hints to literature in respect to topics touched within the overall discussion and to developments that the subject has witnessed in the years, complete the monograph. \par The book can be generally addressed to graduate students and young researchers in physics, theoretical chemistry, applied mathematics and electrical engineering, that are looking for a collection of detailed treatments of some significant solvable problems in Quantum Mechanics, and/or for a good arena for concrete practicing on the application of the factorization method to second-order ordinary linear differential problems. In this respect, being already familiar with special functions would be clearly helpful. The book may reveal itself as an interesting tool complementary to standard Quantum Mechanics textbooks, in that instructors might be stimulated to discuss exact mathematical treatments through a ladder operators based approach for problems other than the conventional harmonic oscillator (e.g., the infinitely deep square-well and the Morse potentials); almost any section of Parts III and IV can be selected and read with no loss of continuity once essentials of Part II are understood, in fact. \par Obviously, one may wonder if a different outline could have been considered, in order to avoid Parts consisting in few pages. Yet, owing to the Preface's presentation of the factorization method as a milestone of other approaches, there may be critical readers which, thinking about group theoretical approaches in a wide perspective, could find such a statement not satisfactory elucidated and supported. On the other hand, while factoring linear ordinary operators is a well studied problem, much less is known about factoring linear partial differential operators in several variables (see e.g. [{\it S. P. Tsarev}, On factorization and solution of multidimensional linear partial differential equations, arxiv.org/abs/cs/0609075 (2006); {\it E. S. Shemyakova} and {\it F. Wincler}, Obstacles to factorization of partial differential operators into several factors, Program. Comput. Softw. 33, No. 2, 67--73 (2007; Zbl 1130.35007)], and the recent claim in [{\it R. Beals} and {\it E. A. Kartashova}, Constructive factorization of linear partial differential operators; Theor. Math. Phys. 145, No. 2, 1511--1524 (2005)] about an entirely algebraic procedure for the case of two variables). Nevertheless, these are minor possible comments that are quite far from entering in a real assessment of the whole book's substance and do not detract from general merits of the author in providing a comprehensive self-consistent monograph, where a number of quantum systems of interest are collected and presented in a fashion that complements analytical aspects of the factorization method, as traditionally formulated, with algebraic aspects and results associated with its recent developments.
[Giulio Landolfi (Lecce)]

MSC 2000:: *81-01 Textbooks (quantum theory)
81Q05 Closed and approximate solutions to quantum-mechanical equations
81U15 Exactly and quasi-solvable systems
34L40 Particular ordinary differential operators
81R15 Operator algebra methods
81R05 Repres. of finite-dim. groups and algebras from quantum theory
81Q60 Supersymmetric quantum mechanics
81R30 Coherent states in quantum theory

Keywords: Schrödinger equation; solvable quantum systems; factorization method; ladder operators; dynamic group; coherent states; SUSYQM; shape invariance; control theory

Citations: Zbl 1130.35007

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