## Schrödinger Operators: With Applications to Quantum Mechanics and Global GeometryA complete understanding of Schrödinger operators is a necessary prerequisite for unveiling the physics of nonrelativistic quantum mechanics. Furthermore recent research shows that it also helps to deepen our insight into global differential geometry. This monograph written for both graduate students and researchers summarizes and synthesizes the theory of Schrödinger operators emphasizing the progress made in the last decade by Lieb, Enss, Witten and others. Besides general properties, the book covers, in particular, multiparticle quantum mechanics including bound states of Coulomb systems and scattering theory, quantum mechanics in constant electric and magnetic fields, Schrödinger operators with random and almost periodic potentials and, finally, Schrödinger operator methods in differential geometry to prove the Morse inequalities and the index theorem. This corrected and extended reprint contains updated proofs and references as well as notes on the development in the field over the past twenty years. |

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### Contents

1 SelfAdjointness | 1 |

12 The Classes S and K | 3 |

13 Katos Inequality and All That | 8 |

14 The LeinfelderSimader Theorem | 11 |

2 L𝑃Properties of Eigenfunctions and All That | 13 |

22 Estimates on Eigenfunctions | 17 |

23 Local Estimates on Gradients | 19 |

24 Eigenfunctions and Spectrum Schnols Theorem | 20 |

74 Howlands Formalism and as Floquet Operators | 146 |

75 Potentials and TimeDependent Problems | 149 |

Notes Added for this Reprint | 152 |

8 Complex Scaling | 153 |

82 Translation Analyticity | 158 |

83 Higher Order Mourre Theory | 159 |

84 Computational Aspects of Complex Scaling | 160 |

85 Complex Scaling and the DCStark Effect | 161 |

25 The AllegrettoPiepenbrink Theorem | 22 |

26 Integral Kernals for exptH | 24 |

3 Geometric Methods for Bound States | 27 |

32 Multiparticle Schrödinger Operators | 29 |

33 The HVZTheorem | 32 |

34 More on the Essential Spectrum | 36 |

Widely Separated Bumps | 39 |

A WarmUp | 41 |

37 The RuskaiSigal Theorem | 43 |

38 Liebs Improvement of the RuskaiSigal Theorem | 50 |

39 NBody Systems with Finitely Many Bound States | 52 |

The StoneWeierstrass Gavotte | 58 |

4 Local Commutator Estimates | 60 |

42 Control of Imbedded Eigenvalues | 65 |

43 Absence of Singular Continuous Spectrum | 68 |

44 Exponential Bounds and Nonexistence | 74 |

45 The Mourre Estimate for NBody Schrödinger Operators | 82 |

5 Phase Space Analysis of Scattering | 89 |

52 Perrys Estimate | 92 |

53 Enss Version of Cooks Method | 95 |

54 RAGE Theorems | 97 |

55 Asymptotics of Observables | 101 |

56 Asymptotic Completeness | 105 |

57 Asymptotic Completeness in the ThreeBody Case | 106 |

Notes Added for this Reprint | 114 |

6 Magnetic Fields | 115 |

61 Gauge Invariance and the Essential Spectrum | 117 |

62 A Schrödinger Operator with Dense Point Spectrum | 120 |

63 Supersymmetry in 0Space Dimensions | 121 |

64 The AharonovCasher Result on Zero Energy Eigenstates | 126 |

65 A Theorem of Iwatsuka | 130 |

66 An Introduction to Other Phenomena in Magnetic Fields | 131 |

Notes Added for this Reprint | 134 |

7 Electric Fields | 135 |

72 A Theorem Needed for the Mourre Theory of the OneDimensional Electric Field | 137 |

73 Propagators for TimeDependent Electric Fields | 140 |

86 Complex Scaling and the ACStark Effect | 163 |

87 Extensions and Generalizations | 165 |

Notes Added for this Reprint | 167 |

9 Random Jacobi Matrices | 168 |

91 Basic Deﬁnitions and Results | 169 |

92 The Density of States | 175 |

93 The Lyaponov Exponent and the IshiiPasturKotani Theorem | 180 |

94 Subharmonicity of the Lyaponov Exponent and the Thouless Formula | 186 |

95 Point Spectrum for the Anderson Model | 190 |

Notes Added for this Reprint | 201 |

10 Almost Periodic Jacobi Matrices | 203 |

102 The Almost Mathieu Equation and the Occurrence of Singular Continuous Spectrum | 205 |

103 Pure Point Spectrum and the Maryland Model | 209 |

104 Cantor Sets on Recurrent Absolutely Continuous Spectrum | 218 |

Notes Added for this Reprint | 223 |

11 Wittens Proof of the Morse Inequalities | 224 |

112 The Morse Inequalities | 230 |

113 Hodge Theory | 233 |

114 Wittens Deformed Laplacian | 238 |

115 Proof of Theorem 114 | 241 |

12 Patodis Proof of the GaussBonnetChern Theorem and Superproofs of Index Theorems | 245 |

122 The BerezinPatodi Formula | 254 |

Statement and Strategy of the Proof | 259 |

124 Bochner Laplacian and the Weitzenböck Formula | 265 |

125 Elliptic Regularity | 272 |

126 A Canonical Order Calculus | 280 |

127 Cutting and Pasting | 290 |

128 Completion of the Proof of the GaussBonnetChern Theorem | 291 |

129 Mehlers Formula | 293 |

1210 Introduction to the Index Theorem for Dirac Operators | 304 |

307 | |

321 | |

325 | |

326 | |