This Work introduces the factorization method in quantum mechanics at an advanced level with an aim to put mathematical and physical concepts and techniques like the factorization method, Lie algebras, matrix elements and quantum control at the Reader's disposal. For this purpose a comprehensive description is provided of the factorization method and its wide applications in quantum mechanics which complements the traditional coverage found in the existing quantum mechanics textbooks. Related to this classic method are the supersymmetric quantum mechanics, shape invariant potentials and group theoretical approaches. It is no exaggeration to say that this method has become the milestone of these approaches. In fact the Author's driving force has been his desire to provide a comprehensive review volume that includes some new and significant results about the factorization method in quantum mechanics since the literature is inundated with scattered articles in this field, and to pave the Reader's way into this territory as rapidly as possible. The result: clear and understandable derivations with the necessary mathematical steps included so that the intelligent reader should be able to follow the text with relative ease, in particular when mathematically difficult material is presented.

"This work introduces the factorization method in quantum mechanics at an advanced level with an aim to put mathematical and physical concepts and techniques like the factorization method, Lie algebras, matrix elements and quantum control at the reader's disposal. For this purpose a comprehensive description is provided of the factorization method and its wide applications in quantum mechanics which complements the traditional coverage found in the existing quantum mechanics textbooks. Related to this classic method are the supersymmetric quantum mechanics, shape invariant potentials and group theoretical approaches. It is no exaggeration to say that this method has become the milestone of these approaches. In fact the author's driving force has been his desire to provide a comprehensive review volume that includes some new and significant results about the factorization method in quantum mechanics since the literature is inundated with scattered articles in this field, and to pave the reader's way into this territory as rapidly as possible. The result: clear and understandable derivations with the necessary mathematical steps included so that the intelligent reader should be able to follow the text with relative ease, in particular when mathematically difficult material is presented. TOC:PART I - Introduction. 1: Introduction. PART II - Method. 2: Theory. 3: Lie Algebras SU(2) and SU(1,1). PART III - Applications in Non-Relativistic Quantum mechanics. 4: Harmonic Oscillator. 5: Infinitely Deep Square-Well Potential. 6: Morse Potential. 7: Pöschl-Teller Potential. 8: Pseudoharmonic Oscillator. 9: Algebraic Approach to an Electron in a Uniform Magnetic Field. 10: Ring-Shaped Non-Spherical Oscillator. 11: Generalized Laguerre Functions. 12: New Non-Central Ring-Shaped Potential. 13: Pöschl-Teller Like Potential. 14: Position-Dependent Mass Schrödinger Equation for a Singular Oscillator. PART IV - Applications in Relativistic Quantum Mechanics. 15: SUSYQM and SKWB 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 Dimensions. 17: Algebraic Approach to Klein-Gordon Equation with the Hydrogen-like Atom in 1+2 Dimensions. 18: SUSYQM and SKWB Approaches to Relativistic Dirac and Klein-Gordon Equations with Hyperbolic Potential. PART V - Quantum Control. 19: Controllability of Quantum Systems for the Morse and PT Potentials with Dynamic Group SU(2). 20: Controllability of Quantum System for the PT-like Potential with Dynamic Group SU(1,1). PART VI - Conclusions and Outlooks. 21: Conclusions and outlooks. APPENDICES: A - Integral formulas of the confluent hypergeometric functions. B - Mean values rk for hydrogen-like atom. C - Commutator identities. D - Angular momentum operators in spherical coordinates. E - Confluent hypergeometric function. References. Index."

PART I - Introduction. 1: Introduction. 1.1 Basic review. 1.2. Motivations and aims.PART II - Method. 2: Theory. 2.1. Introduction. 2.2. Formalism. 3: Lie Algebras SU(2) and SU(1,1). 3.1. Introduction. 3.2. Abstract groups. 3.3. Matrix representation. 3.4. properties of groups SU(2) and SO(3). 3.5. Properties of non-compact groups SO(2,1) and SU(1,1). 3.6. Generators of Lie groups SU(2) and SU(1,1). 3.7. Irreducible representations. 3.8. Irreducible unitary representations. 3.9. Concluding remarks.PART III - Applications in Non-Relativistic Quantum mechanics. 4: Harmonic Oscillator. 4.1. Introduction. 4.2. Exact solutions. 4.3. Ladder operators. 4.4. Bargmann-Segal transformations. 4.5. Single mode realization of dynamic group SU(1,1). 4.6. Matrix elements. 4.7. Coherent states. 4.8. Franck-Condon factors. 4.9. Concluding remarks. 5: Infinitely Deep Square-Well Potential. 5.1. Introduction. 5.2. Ladder operators for infinitely deep square-well potential. 5.3. Realization of dynamic group SU(1,1) and matrix elements. 5.4. Ladder operators for infinitely deep symmetric well potential. 5.5. SUSYQM approach to infinitely deep square-well potential. 5.6. Perelomov coherent states. 5.7. Barut-Girardello coherent states. 5.8. Concluding remarks. 6: Morse Potential. 6.1. Introduction. 6.2. Exact solutions. 6.3. Ladder operators for the Morse potential. 6.4. Realization of dynamic group SU(2). 6.5. Matrix elements. 6.6. Harmonic limit. 6.7. Franck-Condon factors. 6.8. Transition probability. 6.9. Realization of dynamic group SU(1,1). 6.10. Concluding remarks. 7: Pöschl-Teller Potential. 7.1. Introduction. 7.2. Exact solutions. 7.3. Ladder operators. 7.4. Realization of dynamic group SU(2). 7.5. Alternative approach to derive ladder operators. 7.6. Harmonic limit. 7.7. Expansions of the coordinate x and momentum p from the SU(2) generators.7.8. Concluding remarks. 8: Pseudoharmonic Oscillator. 8.1. Introduction. 8.2. Exact solutions in one dimension. 8.3. Ladder operators. 8.4. Barut-Girardello coherent states. 8.5. Thermodynamic properties. 8.6. Pseudoharmonic oscillator in arbitrary dimensions. 8.7. Recurrence relations among matrix elements. 8.8. Concluding remarks. 9: Algebraic Approach to an Electron in a Uniform Magnetic Field. 9.1. Introduction. 9.2. Exact solutions. 9.3. Ladder operators. 9.4. Concluding remarks. 10: Ring-Shaped Non-Spherical Oscillator. 10.1. Introduction. 10.2. Exact solutions. 10.3. Ladder operators. 10.4. Realization of dynamic group. 10.5. Concluding remarks. 11: Generalized Laguerre Functions. 11.1. Introduction. 11.2. generalized Laguerre functions. 11.3. Ladder operators and realization of dynamic group SU(1,1). 11.4. Concluding remarks. 12: New Non-Central Ring-Shaped Potential. 12.1. Introduction. 12.2. Bound states. 12.3. Ladder operators. 12.4. Mean values. 12.5. Continuum states. 12.6. Concluding remarks. 13: Pöschl-Teller Like Potential. 13.1. Introduction. 13.2. Exact solutions. 13.3. Ladder operators. 13.4. Realization of dynamic group and matrix elements. 13.5. Infinitely square-well and harmonic limits. 13.6. Concluding remarks. 14: Position-Dependent Mass Schrödinger Equation for a Singular Oscillator. 14.1. Introduction. 14.2. Position-dependent effective mass Schrödinger equation for harmonic oscillator. 14.3. Singular oscillator with a position-dependent effective mass. 14.4. Complete solutions. 14.5. Another position-dependent effective mass. 14.6. Concluding remarks.PART IV - Applications in Relativistic Quantum Mechanics. 15: SUSYQM and SKWB Approach to the Dirac Equation with a Coulomb Potential in 2+1 Dimensions. 15.1. Introduction. 15.2. Dirac equation in 2+1 dimensions. 15.3. Exact solutions. 15.4. SUSYQM