Quantum mechanics and the theory of operators on Hilbert space have been deeply linked since their beginnings in the early twentieth century. States of a quantum system correspond to certain elements of the configuration space and observables correspond to certain operators on the space. This book is a brief, but self-contained, introduction to the mathematical methods of quantum mechanics, with a view towards applications to Schrödinger operators.

Part 1 of the book is a concise introduction to the spectral theory of unbounded operators. Only those topics that will be needed for later applications are covered. The spectral theorem is a central topic in this approach and is introduced at an early stage. Part 2 starts with the free Schrödinger equation and computes the free resolvent and time evolution. Position, momentum, and angular momentum are discussed via algebraic methods. Various mathematical methods are developed, which are then used to compute the spectrum of the hydrogen atom. Further topics include the nondegeneracy of the ground state, spectra of atoms, and scattering theory.

This book serves as a self-contained introduction to spectral theory of unbounded operators in Hilbert space with full proofs and minimal prerequisites: Only a solid knowledge of advanced calculus and a one-semester introduction to complex analysis are required. In particular, no functional analysis and no Lebesgue integration theory are assumed. It develops the mathematical tools necessary to prove some key results in nonrelativistic quantum mechanics.

Mathematical Methods in Quantum Mechanics is intended for beginning graduate students in both mathematics and physics and provides a solid foundation for reading more advanced books and current research literature.

This new edition has additions and improvements throughout the book to make the presentation more student friendly.

Solutions are available electronically for instructors only. Please send email to textbooks@ams.org for more information. Updates and corrections are available for the book; please see errata.

The book is written in a very clear and compact style. It is well suited for self-study and includes numerous exercises (many with hints).

—Zentralblatt MATH

The author presents this material in a very clear and detailed way and supplements it by numerous exercises. This makes the book a nice introduction to this exciting field of mathematics.

—Mathematical Reviews

Readership

Graduate students and research mathematicians interested in spectral theory and quantum mechanics, with an emphasis on Schrödinger operators.

• N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, Vols. I and II, Pitman, Boston, 1981.
• S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden, Solvable Models in Quantum Mechanics, 2nd ed., American Mathematical Society, Providence, 2005.
• W. O. Amrein, Non-Relativistic Quantum Dynamics, D. Reidel, Dordrecht, 1981.
• W. O. Amrein, A. M. Hinz, and D. B. Pearson, Sturm–Liouville Theory: Past and Present, Birkhäuser, Basel, 2005.
• W. O. Amrein, J. M. Jauch, and K. B. Sinha, Scattering Theory in Quantum Mechanics, W. A. Benajmin Inc., New York, 1977.
• V. G. Bagrov and D. M. Gitman, Exact Solutions of Relativistic Wave Equations, Kluwer Academic Publishers, Dordrecht, 1990.
• H. Baumgaertel and M. Wollenberg, Mathematical Scattering Theory, Birkhäuser, Basel, 1983.
• H. Bauer, Measure and Integration Theory, de Gruyter, Berlin, 2001.
• C. Bennewitz, A proof of the local Borg–Marchenko theorem, Commun. Math. Phys. 218, 131–132 (2001).
• A. M. Berthier, Spectral Theory and Wave Operators for the Schrödinger Equation, Pitman, Boston, 1982.
• J. Blank, P. Exner, and M. Havlíček, Hilbert-Space Operators in Quantum Physics, 2nd ed., Springer, Dordrecht, 2008.
• V. I. Bogachev, Measure Theory, 2 vols., Springer, Berlin, 2007.
• R. Carmona and J. Lacroix, Spectral Theory of Random Schrödinger Operators, Birkhäuser, Boston, 1990.
• K. Chadan and P. C. Sabatier, Inverse Problems in Quantum Scattering Theory, Springer, New York, 1989.
• E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, Krieger, Malabar, 1985.
• H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon, Schrödinger Operators, 2nd printing, Springer, Berlin, 2008.
• M. Demuth and M. Krishna, Determining Spectra in Quantum Theory, Birkhäuser, Boston, 2005.
• D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Oxford University Press, Oxford, 1987.
• V. Enss, Asymptotic completeness for quantum mechanical potential scattering, Comm. Math. Phys. 61, 285–291 (1978).
• V. Enß, Schrödinger Operators, lecture notes (unpublished).
• L. D. Fadeev and O. A. Yakubovskiĭ, Lectures on Quantum Mechanics for Mathematics Students, Amer. Math. Soc., Providence, 2009.
• S. Flügge, Practical Quantum Mechanics, Springer, Berlin, 1994.
• L. Grafakos, Classical Fourier Analysis, 2nd ed., Springer, New York, 2008.
• I. Gohberg, S. Goldberg, and N. Krupnik, Traces and Determinants of Linear Operators, Birkhäuser, Basel, 2000.
• J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford University Press, Oxford, 1985.
• S. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, 2nd ed., Springer, Berlin, 2011.
• P. R. Halmos, A Hilbert Space Problem Book, 2nd ed., Springer, New York, 1984.
• P. D. Hislop and I. M. Sigal, Introduction to Spectral Theory, Springer, New York, 1996.
• T. Kato, Perturbation Theory for Linear Operators, Springer, New York, 1966.
• A. Komech, Quantum Mechanics: Genesis and Achievements, Springer, Dordrecht, 2013.
• A. Komech and E. Kopylova, Dispersion Decay and Scattering Theory, John Wiley, Hoboken, 2012.
• P. D. Lax, Functional Analysis, Wiley-Interscience, New York, 2002.
• J. L. Kelly, General Topology, Springer, New York, 1955.
• W. Kirsch, An invitation to random Schrödinger operators, in Random Schrödinger Operators, M. Dissertori et al. (eds.), 1–119, Panoramas et Synthèses 25, Société Mathématique de France, Paris, 2008.
• Y. Last, Quantum dynamics and decompositions of singular continuous spectra, J. Funct. Anal. 142, 406–445 (1996).
• B. M. Levitan, Inverse Sturm–Liouville Problems, VNU Science Press, Utrecht, 1987.
• B. M. Levitan and I. S. Sargsjan, Introduction to Spectral Theory, American Mathematical Society, Providence, 1975.
• B. M. Levitan and I. S. Sargsjan, Sturm–Liouville and Dirac Operators, Kluwer Academic Publishers, Dordrecht, 1991.
• E. Lieb and M. Loss, Analysis, American Mathematical Society, Providence, 1997.
• V. A. Marchenko, Sturm–Liouville Operators and Applications, Birkhäuser, Basel, 1986.
• E. H. Lieb and R. Seiringer, Stability of Matter, Cambridge University Press, Cambridge, 2010.
• M. A. Naimark, Linear Differential Operators, Parts I and II , Ungar, New York, 1967 and 1968.
• R. G. Newton, Scattering Theory of Waves and Particles, 2nd ed., Dover, New York, 2002.
• F. W. J. Olver et al., NIST Handbook of Mathematical Functions, Cambridge University Press, Cambridge, 2010.
• L. Pastur and A. Figotin, Spectra of Random and Almost-Periodic Operators, Springer, Berlin, 1992.
• D. Pearson, Quantum Scattering and Spectral Theory, Academic Press, London, 1988.
• P. Perry, Mellin transforms and scattering theory, Duke Math. J. 47, 187–193 (1987).
• E. Prugovečki, Quantum Mechanics in Hilbert Space, 2nd ed., Academic Press, New York, 1981.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics I. Functional Analysis, rev. and enl. ed., Academic Press, San Diego, 1980.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics II. Fourier Analysis, Self-Adjointness, Academic Press, San Diego, 1975.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics III. Scattering Theory, Academic Press, San Diego, 1979.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics IV. Analysis of Operators, Academic Press, San Diego, 1978.
• J. R. Retherford, Hilbert Space: Compact Operators and the Trace Theorem, Cambridge University Press, Cambridge, 1993.
• G. Roepstorff, Path Integral Approach to Quantum Physics, Springer, Berlin, 1994.
• F. S. Rofe-Beketov and A. M. Kholkin, Spectral Analysis of Differential Operators. Interplay Between Spectral and Oscillatory Properties, World Scientific, Hackensack, 2005.
• W. Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hill, New York, 1987.
• M. Schechter, Operator Methods in Quantum Mechanics, North Holland, New York, 1981.
• B. Simon, Quantum Mechanics for Hamiltonians Defined as Quadratic Forms, Princeton University Press, Princeton, 1971.
• B. Simon, Functional Integration and Quantum Physics, Academic Press, New York, 1979.
• B. Simon, Schrödinger operators in the twentieth century, J. Math. Phys. 41:6, 3523–3555 (2000).
• B. Simon, Trace Ideals and Their Applications, 2nd ed., Amererican Mathematical Society, Providence, 2005.
• E. Stein and R. Shakarchi, Complex Analysis, Princeton University Press, Princeton, 2003.
• L. A. Takhtajan, Quantum Mechanics for Mathematicians, Amer. Math. Soc., Providence, 2008.
• G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices, Math. Surv. and Mon. 72, Amer. Math. Soc., Rhode Island, 2000.
• B. Thaller, The Dirac Equation, Springer, Berlin 1992.
• B. Thaller, Visual Quantum Mechanics, Springer, New York, 2000.
• B. Thaller, Advanced Visual Quantum Mechanics, Springer, New York, 2005.
• W. Thirring, Quantum Mechanics of Atoms and Molecules, Springer, New York, 1981.
• G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge University Press, Cambridge, 1962.
• J. Weidmann, Linear Operators in Hilbert Spaces, Springer, New York, 1980.
• J. Weidmann, Spectral Theory of Ordinary Differential Operators, Lecture Notes in Mathematics, 1258, Springer, Berlin, 1987.
• J. Weidmann, Lineare Operatoren in Hilberträumen, Teil 1: Grundlagen, B. G. Teubner, Stuttgart, 2000.
• J. Weidmann, Lineare Operatoren in Hilberträumen, Teil 2: Anwendungen, B. G. Teubner, Stuttgart, 2003.
• J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, 1996.
• D. R. Yafaev, Mathematical Scattering Theory: General Theory, American Mathematical Society, Providence, 1992.
• K. Yosida, Functional Analysis, 6th ed., Springer, Berlin, 1980.
• A. Zettl, Sturm–Liouville Theory, American Mathematical Society, Providence, 2005.

• N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, Vols. I and II, Pitman, Boston, 1981.
• S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden, Solvable Models in Quantum Mechanics, 2nd ed., American Mathematical Society, Providence, 2005.
• W. O. Amrein, Non-Relativistic Quantum Dynamics, D. Reidel, Dordrecht, 1981.
• W. O. Amrein, A. M. Hinz, and D. B. Pearson, Sturm–Liouville Theory: Past and Present, Birkhäuser, Basel, 2005.
• W. O. Amrein, J. M. Jauch, and K. B. Sinha, Scattering Theory in Quantum Mechanics, W. A. Benajmin Inc., New York, 1977.
• V. G. Bagrov and D. M. Gitman, Exact Solutions of Relativistic Wave Equations, Kluwer Academic Publishers, Dordrecht, 1990.
• H. Baumgaertel and M. Wollenberg, Mathematical Scattering Theory, Birkhäuser, Basel, 1983.
• H. Bauer, Measure and Integration Theory, de Gruyter, Berlin, 2001.
• C. Bennewitz, A proof of the local Borg–Marchenko theorem, Commun. Math. Phys. 218, 131–132 (2001).
• A. M. Berthier, Spectral Theory and Wave Operators for the Schrödinger Equation, Pitman, Boston, 1982.
• J. Blank, P. Exner, and M. Havlíček, Hilbert-Space Operators in Quantum Physics, 2nd ed., Springer, Dordrecht, 2008.
• V. I. Bogachev, Measure Theory, 2 vols., Springer, Berlin, 2007.
• R. Carmona and J. Lacroix, Spectral Theory of Random Schrödinger Operators, Birkhäuser, Boston, 1990.
• K. Chadan and P. C. Sabatier, Inverse Problems in Quantum Scattering Theory, Springer, New York, 1989.
• E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, Krieger, Malabar, 1985.
• H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon, Schrödinger Operators, 2nd printing, Springer, Berlin, 2008.
• M. Demuth and M. Krishna, Determining Spectra in Quantum Theory, Birkhäuser, Boston, 2005.
• D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Oxford University Press, Oxford, 1987.
• V. Enss, Asymptotic completeness for quantum mechanical potential scattering, Comm. Math. Phys. 61, 285–291 (1978).
• V. Enß, Schrödinger Operators, lecture notes (unpublished).
• L. D. Fadeev and O. A. Yakubovskiĭ, Lectures on Quantum Mechanics for Mathematics Students, Amer. Math. Soc., Providence, 2009.
• S. Flügge, Practical Quantum Mechanics, Springer, Berlin, 1994.
• L. Grafakos, Classical Fourier Analysis, 2nd ed., Springer, New York, 2008.
• I. Gohberg, S. Goldberg, and N. Krupnik, Traces and Determinants of Linear Operators, Birkhäuser, Basel, 2000.
• J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford University Press, Oxford, 1985.
• S. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, 2nd ed., Springer, Berlin, 2011.
• P. R. Halmos, A Hilbert Space Problem Book, 2nd ed., Springer, New York, 1984.
• P. D. Hislop and I. M. Sigal, Introduction to Spectral Theory, Springer, New York, 1996.
• T. Kato, Perturbation Theory for Linear Operators, Springer, New York, 1966.
• A. Komech, Quantum Mechanics: Genesis and Achievements, Springer, Dordrecht, 2013.
• A. Komech and E. Kopylova, Dispersion Decay and Scattering Theory, John Wiley, Hoboken, 2012.
• P. D. Lax, Functional Analysis, Wiley-Interscience, New York, 2002.
• J. L. Kelly, General Topology, Springer, New York, 1955.
• W. Kirsch, An invitation to random Schrödinger operators, in Random Schrödinger Operators, M. Dissertori et al. (eds.), 1–119, Panoramas et Synthèses 25, Société Mathématique de France, Paris, 2008.
• Y. Last, Quantum dynamics and decompositions of singular continuous spectra, J. Funct. Anal. 142, 406–445 (1996).
• B. M. Levitan, Inverse Sturm–Liouville Problems, VNU Science Press, Utrecht, 1987.
• B. M. Levitan and I. S. Sargsjan, Introduction to Spectral Theory, American Mathematical Society, Providence, 1975.
• B. M. Levitan and I. S. Sargsjan, Sturm–Liouville and Dirac Operators, Kluwer Academic Publishers, Dordrecht, 1991.
• E. Lieb and M. Loss, Analysis, American Mathematical Society, Providence, 1997.
• V. A. Marchenko, Sturm–Liouville Operators and Applications, Birkhäuser, Basel, 1986.
• E. H. Lieb and R. Seiringer, Stability of Matter, Cambridge University Press, Cambridge, 2010.
• M. A. Naimark, Linear Differential Operators, Parts I and II , Ungar, New York, 1967 and 1968.
• R. G. Newton, Scattering Theory of Waves and Particles, 2nd ed., Dover, New York, 2002.
• F. W. J. Olver et al., NIST Handbook of Mathematical Functions, Cambridge University Press, Cambridge, 2010.
• L. Pastur and A. Figotin, Spectra of Random and Almost-Periodic Operators, Springer, Berlin, 1992.
• D. Pearson, Quantum Scattering and Spectral Theory, Academic Press, London, 1988.
• P. Perry, Mellin transforms and scattering theory, Duke Math. J. 47, 187–193 (1987).
• E. Prugovečki, Quantum Mechanics in Hilbert Space, 2nd ed., Academic Press, New York, 1981.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics I. Functional Analysis, rev. and enl. ed., Academic Press, San Diego, 1980.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics II. Fourier Analysis, Self-Adjointness, Academic Press, San Diego, 1975.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics III. Scattering Theory, Academic Press, San Diego, 1979.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics IV. Analysis of Operators, Academic Press, San Diego, 1978.
• J. R. Retherford, Hilbert Space: Compact Operators and the Trace Theorem, Cambridge University Press, Cambridge, 1993.
• G. Roepstorff, Path Integral Approach to Quantum Physics, Springer, Berlin, 1994.
• F. S. Rofe-Beketov and A. M. Kholkin, Spectral Analysis of Differential Operators. Interplay Between Spectral and Oscillatory Properties, World Scientific, Hackensack, 2005.
• W. Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hill, New York, 1987.
• M. Schechter, Operator Methods in Quantum Mechanics, North Holland, New York, 1981.
• B. Simon, Quantum Mechanics for Hamiltonians Defined as Quadratic Forms, Princeton University Press, Princeton, 1971.
• B. Simon, Functional Integration and Quantum Physics, Academic Press, New York, 1979.
• B. Simon, Schrödinger operators in the twentieth century, J. Math. Phys. 41:6, 3523–3555 (2000).
• B. Simon, Trace Ideals and Their Applications, 2nd ed., Amererican Mathematical Society, Providence, 2005.
• E. Stein and R. Shakarchi, Complex Analysis, Princeton University Press, Princeton, 2003.
• L. A. Takhtajan, Quantum Mechanics for Mathematicians, Amer. Math. Soc., Providence, 2008.
• G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices, Math. Surv. and Mon. 72, Amer. Math. Soc., Rhode Island, 2000.
• B. Thaller, The Dirac Equation, Springer, Berlin 1992.
• B. Thaller, Visual Quantum Mechanics, Springer, New York, 2000.
• B. Thaller, Advanced Visual Quantum Mechanics, Springer, New York, 2005.
• W. Thirring, Quantum Mechanics of Atoms and Molecules, Springer, New York, 1981.
• G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge University Press, Cambridge, 1962.
• J. Weidmann, Linear Operators in Hilbert Spaces, Springer, New York, 1980.
• J. Weidmann, Spectral Theory of Ordinary Differential Operators, Lecture Notes in Mathematics, 1258, Springer, Berlin, 1987.
• J. Weidmann, Lineare Operatoren in Hilberträumen, Teil 1: Grundlagen, B. G. Teubner, Stuttgart, 2000.
• J. Weidmann, Lineare Operatoren in Hilberträumen, Teil 2: Anwendungen, B. G. Teubner, Stuttgart, 2003.
• J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, 1996.
• D. R. Yafaev, Mathematical Scattering Theory: General Theory, American Mathematical Society, Providence, 1992.
• K. Yosida, Functional Analysis, 6th ed., Springer, Berlin, 1980.
• A. Zettl, Sturm–Liouville Theory, American Mathematical Society, Providence, 2005.