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Über dieses Buch

The prime goal of this monograph, which comprises a total of five volumes, is to derive sharp spectral asymptotics for broad classes of partial differential operators using techniques from semiclassical microlocal analysis, in particular, propagation of singularities, and to subsequently use the variational estimates in “small” domains to consider domains with singularities of different kinds. In turn, the general theory (results and methods developed) is applied to the Magnetic Schrödinger operator, miscellaneous problems, and multiparticle quantum theory.

In this volume the methods developed in Volumes I, II and III are applied to the Schrödinger and Dirac operators in non-smooth settings and in higher dimensions.

Inhaltsverzeichnis

Frontmatter

Non-smooth Theory and Higher Dimensions

Frontmatter

Chapter 18. 2D- and 3D-magnetic Schrödinger Operator with Irregular Coefficients

Abstract
This Chapter is a continuation of Chapter 13 and Sections 4.6 and 5.4. Here we deal with the Schrödinger operator with the strong magnetic field in dimensions 2 and 3 under weak smoothness assumptions.
Victor Ivrii

Chapter 19. Multidimensional Magnetic Schrödinger Operator. Full-Rank Case

Abstract
In this and the next chapters we consider multidimensional Schrödinger Operator.
Victor Ivrii

Chapter 20. Multidimensional Magnetic Schrödinger Operator. Non-Full-Rank Case

Abstract
In this chapter we consider multidimensional Schrödinger operator.
Victor Ivrii

Magnetic Schrödinger Operator in Dimension 4

Frontmatter

Chapter 21. 4D-Schrödinger Operator with a Degenerating Magnetic Field

Abstract
We continue analysis of the Schrödinger operator with the strong degenerating magnetic field, started in Chapter 14.
Victor Ivrii

Chapter 22. Generic 4D-Schrödinger Operator with the Strong Magnetic Field

Abstract
Sharp spectral asymptotics for multidimensional Magnetic Schrödinger were obtained in Chapters 19 and 20 in full- and non-full-rank cases respectively. The results, as one could expect from the analysis of 2- and 3-dimensional cases, were rather different.
Victor Ivrii

Eigenvalue Asymptotics for Schrödinger and Dirac Operators with the Strong Magnetic Field

Frontmatter

Chapter 23. Eigenvalue Asymptotics. 2D Case

Abstract
In this chapter we obtain eigenvalue asymptotics for 2D-Schrödinger, Schrödinger-Pauli and Dirac operators in the situations in which the role of the magnetic field is important. We have seen in Chapters 13 and 17 that these operators are essentially different and they also differ significantly from the corresponding 3D-operators.
Victor Ivrii

Chapter 24. Eigenvalue asymptotics. 3D case

Abstract
In this chapter we obtain eigenvalue asymptotics for 3D-Schrödinger, Schrödinger-Pauli and Dirac operators in the situations in which the role of the magnetic field is important.
Victor Ivrii

Backmatter

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