Semilinear Schrödinger equations / Thierry Cazenave.

Format:

Book

Author/Creator:

Cazenave, Thierry.

Series:

Courant lecture notes in mathematics ; 10.

Courant lecture notes ; 10

Language:

English

Subjects (All):

Schrödinger equation.

Physical Description:

xiii, 323 pages : illustrations ; 26 cm.

Place of Publication:

New York : Courant Institute of Mathematical Sciences ; Providence, R.I. : American Mathematical Society, [2003]

Contents:

1. Preliminary Results 1

1.1. Functional Analysis 1

1.2. Integration 3

1.3. Sobolev Spaces 6

1.4. Sobolev and Besov Spaces on R[superscript N] 12

1.5. Elliptic Equations 15

1.6. Semigroups of Linear Operators 18

1.7. Some Compactness Tools 20

2. The Linear Schrodinger Equation 29

2.1. Basic Properties 29

2.2. Fundamental Properties in R[superscript N] 30

2.3. Strichartz's Estimates 33

2.4. Strichartz's Estimates for Nonadmissible Pairs 40

2.5. Space Decay and Smoothing Effect in R[superscript N] 41

2.6. Homogeneous Data in R[superscript N] 43

3. The Cauchy Problem in a General Domain 55

3.1. The Notion of Solution 55

3.2. Some Typical Nonlinearities 58

3.3. Local Existence in the Energy Space 63

3.4. Energy Estimates and Global Existence 74

3.5. The Nonlinear Schrodinger Equation in One Dimension 77

3.6. The Nonlinear Schrodinger Equation in Two Dimensions 78

4. The Local Cauchy Problem 83

4.2. Strichartz's Estimates and Uniqueness 84

4.3. Local Existence in H[superscript 1] (R[superscript N]) 92

4.4. Kato's Method 93

4.5. A Critical Case in H[superscript 1] (R[superscript N]) 103

4.6. L[superscript 2] Solutions 109

4.7. A Critical Case in L[superscript 2] (R[superscript N]) 115

4.8. H[superscript 2] Solutions 119

4.9. H[superscript s] Solutions, s [less than sign] N/2 125

4.10. H[superscript m] Solutions, m [greater than sign] N/2 134

4.11. The Cauchy Problem for a Nonautonomous Schrodinger Equation 138

5. Regularity and the Smoothing Effect 147

5.1. H[superscript s] Regularity, 0 [less than sign] s [less than sign] min{1, N/2} 147

5.2. H[superscript 1] Regularity 149

5.3. H[superscript 2] Regularity 152

5.4. H[superscript m] Regularity, m [greater than sign] N/2 154

5.5. Arbitrary Regularity 155

5.6. The C[superscript infinity] Smoothing Effect 156

6. Global Existence and Finite-Time Blowup 163

6.1. Energy estimates and Global Existence 163

6.2. Global Existence for Small Data 165

6.3. Global Existence for Oscillating Data 167

6.4. Global Existence for Asymptotically Homogeneous Initial Data 171

6.5. Finite-Time Blowup 178

6.6. The Critical Case: Sharp Existence and Blowup Results 196

6.7. The Pseudoconformal Transformation and Applications 204

7. Asymptotic Behavior in the Repulsive Case 211

7.1. Basic Notions of Scattering Theory 211

7.2. The Pseudoconformal Conservation Law 213

7.3. Decay of Solutions in the Weighted L[superscript 2] Space 215

7.4. Scattering Theory in the Weighted L[superscript 2] Space 220

7.5. Applications of the Pseudoconformal Transformation 224

7.6. Morawetz's Estimate 233

7.7. Decay of Solutions in the Energy Space 239

7.8. Scattering Theory in the Energy Space 246

8. Stability of Bound States in the Attractive Case 255

8.1. Nonlinear Bound States 255

8.2. An Instability Result 269

8.3. A Stability Result 274

9. Further Results 283

9.1. The Nonlinear Schrodinger Equation with a Magnetic Field 283

9.2. The nonlinear Schrodinger Equation with a Quadratic Potential 287

9.3. The Logarithmic Schrodinger Equation 291

9.4. Existence of Weak Solutions for Large Nonlinearities 299.

Notes:

Includes bibliographical references (pages 305-323)

ISBN:

0821833995

OCLC:

52554437