The d-bar Neumann Problem and Schrödinger Operators (eBook)

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Friedrich Haslinger,University of Vienna, Austria.

Preface 7
Contents 9
1 Bergman spaces 13
1.1 Elementary properties 13
1.2 Examples 20
1.3 Biholomorphic maps 24
1.4 Notes 27
2 The canonical solution operator to ?? 28
2.1 Compact operators on Hilbert spaces 28
2.2 The canonical solution operator to 
38 
2.3 Notes 47
3 Spectral properties of the canonical solution operator to 
48 
3.1 Complex differential forms 48
3.2 (0, 1)-forms with holomorphic coefficients 49
3.3 Compactness and Schatten class membership 51
3.4 Notes 61
4 The .¯ -complex 
62 
4.1 Unbounded operators on Hilbert spaces 62
4.2 Distributions 77
4.3 A finite-dimensional analog 83
4.4 The .¯ -Neumann operator 
84 
4.5 Notes 98
5 Density of smooth forms 99
5.1 Friedrichs’ Lemma and Sobolev spaces 99
5.2 Density in the graph norm 110
5.3 Notes 115
6 The weighted .¯-complex 116
6.1 The .¯-Neumann operator 
116 
6.2 (0, q)-forms 
121 
6.3 Notes 124
7 The twisted .¯-complex 
126 
7.1 An exact sequence of unbounded operators 126
7.2 The twisted basic estimates 127
7.3 Notes 130
8 Applications 131
8.1 Hörmander’s L2-estimates 
131 
8.2 Weighted spaces of entire functions 134
8.3 Notes 138
9 Spectral analysis 139
9.1 Resolutions of the identity 139
9.2 Spectral decomposition of bounded normal operators 142
9.3 Spectral decomposition of unbounded self-adjoint operators 148
9.4 Determination of the spectrum 162
9.5 Variational characterization of the discrete spectrum 172
9.6 Notes 177
10 Schrödinger operators and Witten–Laplacians 178
10.1 Difference quotients 178
10.2 Interior regularity 180
10.3 Schrödinger operators with magnetic field 183
10.4 Witten–Laplacians 190
10.5 Dirac and Pauli operators 192
10.6 Notes 194
11 Compactness 195
11.1 Precompact sets in L2-spaces 
195 
11.2 Sobolev spaces and Gårding’s inequality 198
11.3 Compactness in weighted spaces 202
11.4 Bounded pseudoconvex domains 214
11.5 Notes 217
12 The .¯-Neumann 
219 
12.1 The Stone–Weierstraß Theorem 219
12.2 Commutators of the Bergman projection 222
12.3 Notes 227
13 Compact resolvents 228
13.1 Schrödinger operators 228
13.2 Dirac and Pauli operators 230
13.3 Notes 232
14 Spectrum of . 
233 
14.1 The general setting 233
14.2 Determination of the spectrum 235
14.3 Notes 240
15 Obstructions to compactness 241
15.1 The bidisc 241
15.2 Weighted spaces 242
15.3 Notes 246
Bibliography 247
Index 251

lt;P>"This monograph is a valuable introduction to an active area of contemporary research at the intersection of multidimensional complex analysis, operator theory, and partial differential equations." Zentralblatt für Mathematik