**Contemporary Mathematics**

Volume: 640;
2015;
197 pp;
Softcover

MSC: Primary 35;

**Print ISBN: 978-1-4704-0989-0
Product Code: CONM/640**

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**Electronic ISBN: 978-1-4704-2493-0
Product Code: CONM/640.E**

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# Spectral Theory and Partial Differential Equations

Share this page *Edited by *
*Gregory Eskin; Leonid Friedlander; John Garnett*

This volume contains the proceedings of the
Conference on Spectral Theory and Partial Differential Equations, held
from June 17–21, 2013, at the University of California, Los Angeles,
California, in honor of James Ralston's 70th Birthday.

Papers in this volume cover important topics in spectral theory and
partial differential equations such as inverse problems, both
analytical and algebraic; minimal partitions and Pleijel's Theorem;
spectral theory for a model in Quantum Field Theory; and beams on Zoll
manifolds.

#### Readership

Graduate students and research mathematicians interested in spectral theory, mathematical physics, inverse problems, analysis, PDEs, and functional analysis.

# Table of Contents

## Spectral Theory and Partial Differential Equations

- Cover Cover11 free
- Title page iii4 free
- Contents vii8 free
- Preface ix10 free
- Algebras in reconstruction of manifolds 112 free
- Spectral theory of a mathematical model in quantum field theory for any spin 1324
- 1. Introduction 1324
- 2. Free causal fields for massive and massless particles of any spin 1425
- 3. Definition of the model 1930
- 4. A self-adjoint Hamiltonian 2233
- 5. Main results 2435
- 6. Spectral gap for cutoff operators 2637
- 7. Proof of theorem 5.2 2839
- 8. Proof of theorem 5.3 3041
- Acknowledgements 3546
- References 3546

- A review on large 𝑘 minimal spectral 𝑘-partitions and Pleijel’s Theorem 3950
- 1. Introduction 3950
- 2. A reminder on minimal spectral partitions 4051
- 3. Pleijel’s Theorem revisited 4152
- 4. Improving the use of the Faber-Krahn Inequality 4556
- 5. Considerations around rectangles 4859
- 6. Looking for a class 𝒪^{#}. 5061
- 7. Pleijel’s Theorem for Aharonov-Bohm operators and application to minimal partitions 5364
- References 5667

- Increasing stability for near field from the scattering amplitude 5970
- Inverse scattering on multi-dimensional asymptotically hyperbolic orbifolds 7182
- Error estimates of the Bloch band-based Gaussian beam superposition for the Schrödinger equation 8798
- On random weighted Sobolev inequalities on ℝ^{𝕕} and applications 115126
- 1. Introduction 115126
- 2. Spectral estimates and harmonic Sobolev spaces 116127
- 3. Probabilistic weighted estimates for frequency localized functions 118129
- 4. Hermite functions estimates 121132
- 5. Application to quantum ergodicity 123134
- 6. Application to supercritical nonlinear Schrödinger equations 125136
- Appendix A. Proof of Theorem 3.2 131142
- Appendix B. Proof of Theorem 3.5 132143
- References 133144

- Calderón problem for Maxwell’s equations in the waveguide 137148
- 1. Introduction and main results 137148
- 2. Step 1: Reduction of the Maxwell’s equations to the decoupled system of elliptic equations 139150
- 3. Step 2: Construction of the operators 𝑃_{𝐵} and 𝑇_{𝐵} 142153
- 4. Step 3: Construction of complex geometric optics solutions 145156
- 5. Step 4: Asymptotics 156167
- 6. Step 5: End of the proof 163174
- References 168179

- Gaussian beams on Zoll manifolds and maximally degenerate Laplacians 169180
- Back Cover Back Cover1210