20世纪初，量子力学和Hilbert空间上的算子理论已密切相关。量子系统的状态对应于位形空间的特定元素，可观测量对应于空间上的特定算子。本书是对量子力学数学方法的一个简要但自封的介绍，着眼于Schr?dinger算子的应用。 第一部分简要介绍无界算子的谱理论，仅讨论后面应用所需的内容。谱定理是这种方法的核心，在开篇就会介绍。第二部分从自由Schr?dinger方程开始，计算自由预解式和时间演化；位置、动量和角动量将用代数方法讨论；详尽介绍了各种数学方法，然后将其用于计算氢原子的光谱。进一步的主题包括基态的非简并性、原子光谱和散射理论。 本书是关于Hilbert空间中无界算子谱理论的一个自封的介绍，提供了完整的证明和最少的预备知识——仅要求读者有扎实的高等微积分和一学期复分析导论的知识。特别地，本书不要求读者有泛函分析和Lebesgue积分理论的知识。它介绍了必要的数学工具来证明非相对论量子力学的一些关键结果。 本书面向数学和物理专业的低年级研究生，为他们阅读更高级的图书和当前研究文献奠定坚实基础。 第二版对整本书进行了增补和改进，更便于学生阅读。

前辅文
Preface
Part 0. Preliminaries
  Chapter 0. A first look at Banach and Hilbert spaces
   0.1. Warm up: Metric and topological spaces
   0.2. The Banach space of continuous functions
   0.3. The geometry of Hilbert spaces
   0.4. Completeness
   0.5. Bounded operators
   0.6. Lebesgue Lp spaces
   0.7. Appendix: The uniform boundedness principle
Part 1. Mathematical Foundations of Quantum Mechanics
  Chapter 1. Hilbert spaces
   1.1. Hilbert spaces
   1.2. Orthonormal bases
   1.3. The projection theorem and the Riesz lemma
   1.4. Orthogonal sums and tensor products
   1.5. The C* algebra of bounded linear operators
   1.6. Weak and strong convergence
   1.7. Appendix: The Stone–Weierstraß theorem
  Chapter 2. Self-adjointness and spectrum
   2.1. Some quantum mechanics
   2.2. Self-adjoint operators
   2.3. Quadratic forms and the Friedrichs extension
   2.4. Resolvents and spectra
   2.5. Orthogonal sums of operators
   2.6. Self-adjoint extensions
   2.7. Appendix: Absolutely continuous functions
  Chapter 3. The spectral theorem
   3.1. The spectral theorem
   3.2. More on Borel measures
   3.3. Spectral types
   3.4. Appendix: Herglotz–Nevanlinna functions
  Chapter 4. Applications of the spectral theorem
   4.1. Integral formulas
   4.2. Commuting operators
   4.3. Polar decomposition
   4.4. The min-max theorem
   4.5. Estimating eigenspaces
   4.6. Tensor products of operators
  Chapter 5. Quantum dynamics
   5.1. The time evolution and Stone’s theorem
   5.2. The RAGE theorem
   5.3. The Trotter product formula
  Chapter 6. Perturbation theory for self-adjoint operators
   6.1. Relatively bounded operators and the Kato–Rellich theorem
   6.2. More on compact operators
   6.3. Hilbert–Schmidt and trace class operators
   6.4. Relatively compact operators and Weyl’s theorem
   6.5. Relatively form-bounded operators and the KLMN theorem
   6.6. Strong and norm resolvent convergence
Part 2. Schrödinger Operators
  Chapter 7. The free Schrödinger operator
   7.1. The Fourier transform
   7.2. Sobolev spaces
   7.3. The free Schrödinger operator
   7.4. The time evolution in the free case
   7.5. The resolvent and Green’s function
  Chapter 8. Algebraic methods
   8.1. Position and momentum
   8.2. Angular momentum
   8.3. The harmonic oscillator
   8.4. Abstract commutation
  Chapter 9. One-dimensional Schrödinger operators
   9.1. Sturm–Liouville operators
   9.2. Weyl’s limit circle, limit point alternative
   9.3. Spectral transformations I
   9.4. Inverse spectral theory
   9.5. Absolutely continuous spectrum
   9.6. Spectral transformations II
   9.7. The spectra of one-dimensional Schrödinger operators
  Chapter 10. One-particle Schrödinger operators
   10.1. Self-adjointness and spectrum
   10.2. The hydrogen atom
   10.3. Angular momentum
   10.4. The eigenvalues of the hydrogen atom
   10.5. Nondegeneracy of the ground state
  Chapter 11. Atomic Schrödinger operators
   11.1. Self-adjointness
   11.2. The HVZ theorem
  Chapter 12. Scattering theory
   12.1. Abstract theory
   12.2. Incoming and outgoing states
   12.3. Schrödinger operators with short range potentials
Part 3. Appendix
  Appendix A. Almost everything about Lebesgue integration
   A.1. Borel measures in a nutshell
   A.2. Extending a premeasure to a measure
   A.3. Measurable functions
   A.4. How wild are measurable objects?
   A.5. Integration—Sum me up, Henri
   A.6. Product measures
   A.7. Transformation of measures and integrals
   A.8. Vague convergence of measures
   A.9. Decomposition of measures
   A.10. Derivatives of measures
Bibliographical notes
Bibliography
Glossary of notation
Index