Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top

2019 | Book

No Magnetic Field Case Read chapter Read first chapter

Microlocal Analysis, Sharp Spectral Asymptotics and Applications V

Applications to Quantum Theory and Miscellaneous Problems

Author: Prof. Victor Ivrii

Publisher: Springer International Publishing

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Login to get access
insite
SEARCH

About this book

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, III and IV are applied to multiparticle quantum theory (asymptotics of the ground state energy and related problems), and to miscellaneous spectral problems.

Table of Contents

Frontmatter

Application to Multiparticle Quantum Theory

Frontmatter
Chapter 25. No Magnetic Field Case
Abstract
The purpose of this Part is to apply semiclassical methods developed in the previous parts to the theory of heavy atoms and molecules. Because of this we combine our semiclassical methods with the traditional methods of that theory, mainly function-analytic.
Victor Ivrii
Chapter 26. The Case of External Magnetic Field
Abstract
In this Chapter we repeat analysis of the previous Chapter 25 but in the case of the constant external magnetic field
Victor Ivrii
Chapter 27. The Case of Self-Generated Magnetic Field
Abstract
We are going to replace Schrödinger operator without magnetic field as in Chapter  or with a constant magnetic field as in Chapter  by Schrödinger operator
Victor Ivrii
Chapter 28. The Case of Combined Magnetic Field
Abstract
In this Chapter instead of the Schrödinger operator without magnetic field as in Chapter 25, or with a constant magnetic field as in Chapter 26, or with a self-generated magnetic field as in Chapter 27
Victor Ivrii
Backmatter

Articles

Frontmatter
Chapter 29. Spectral Asymptotics for the Semiclassical Dirichlet to Neumann Operator
Abstract
Let M be a compact Riemannian manifold with smooth boundary, and let \(R(\lambda )\) be the Dirichlet-to-Neumann operator at frequency \(\lambda \). The semiclassical Dirichlet-to-Neumann operator \(R_{\text{scl}}(\lambda) \) is defined to be \(\lambda^{-1} R(\lambda) \). We obtain a leading asymptotic for the spectral counting function for \(R_{\text{scl}}(\lambda) \) in an interval \([a_1, a_2) \) as \(\lambda \to \infty \), under the assumption that the measure of periodic billiards on \(T^*M \) is zero. The asymptotic takes the form
$$\mathsf{N}(\lambda; a_1,a_2) = \bigl( \kappa(a_2)-\kappa(a_1)\bigr)\text{vol}'(\partial M) \lambda^{d-1}+o(\lambda^{d-1}), $$
where \(\kappa(a) \) is given explicitly by
$$\begin{aligned}\kappa(a) &= \frac{\omega_{d-1}}{(2\pi)^{d-1}} \bigg( -\frac{1}{2\pi} \int_{-1}^1 (1 - \eta^2)^{(d-1)/2} \frac{a}{a^2 + \eta^2} \, d\eta \\& \qquad\qquad\qquad\qquad - \frac{1}{4} + H(a) (1+a^2)^{(d-1)/2} \bigg) .\end{aligned} $$
Andrew Hassell, Victor Ivrii
Chapter 30. Spectral Asymptotics for Fractional Laplacians
Abstract
In this article we consider fractional Laplacians which seem to be of interest to probability theory. This is a rather new class of operators for us but our methods works (with a twist, as usual). Our main goal is to derive a two-term asymptotics since one-term asymptotics is easily obtained by R. Seeley’s method.
Victor Ivrii
Chapter 31. Spectral Asymptotics for Dirichlet to Neumann Operator in the Domains with Edges
Abstract
We consider eigenvalues of the Dirichlet-to-Neumann operator for Laplacian in the domain (or manifold) with edges and establish the asymptotics of the eigenvalue counting function
$$N(\lambda )= \kappa _0\lambda ^d +O(\lambda ^{d-1})\qquad \text {as} \lambda \rightarrow +\infty $$
where d is dimension of the boundary. Further, in certain cases we establish two-term asymptotics
$$N(\lambda )=\kappa _0\lambda ^d+\kappa _1\lambda ^{d-1}+o(\lambda ^{d-1})\qquad \text {as} \lambda \rightarrow +\infty $$
We also establish improved asymptotics for Riesz means.
Victor Ivrii
Chapter 32. Asymptotics of the Ground State Energy in the Relativistic Settings
Abstract
The purpose of this paper is to derive sharp asymptotics of the ground state energy for the heavy atoms and molecules in the relativistic settings, and, in particular, to derive relativistic Scott correction term and also Dirac, Schwinger and relativistic correction terms. Also we will prove that Thomas-Fermi density approximates the actual density of the ground state, which opens the way to estimate the excessive negative and positive charges and the ionization energy.
Victor Ivrii
Chapter 33. Asymptotics of the ground state energy in the relativistic settings and with self-generated magnetic field
Abstract
The purpose of this paper is to derive sharp asymptotics of the ground state energy for the heavy atoms and molecules in the relativistic settings, with the self-generated magnetic field, and, in particular, to derive relativistic Scott correction term and also Dirac, Schwinger and relativistic correction terms. Also we will prove that Thomas-Fermi density approximates the actual density of the ground state, which opens the way to estimate the excessive negative and positive charges and the ionization energy.
Andrew Hassell, Victor Ivrii
Chapter 34. Complete Semiclassical Spectral Asymptotics for Periodic and Almost Periodic Perturbations of Constant Operators
Abstract
Under certain assumptions we derive a complete semiclassical asymptotics of the spectral function \(e_{h,\varepsilon }(x, x,\lambda )\) for a scalar operator
$$\begin{aligned} A_\varepsilon (x, hD)= A^0(hD) + \varepsilon B(x, hD), \end{aligned}$$
where \(A^0\) is an elliptic operator and B(x, hD) is a periodic or almost periodic perturbation.
In particular, a complete semiclassical asymptotics of the integrated density of states also holds. Further, we consider generalizations.
Victor Ivrii
Chapter 35. Complete Differentiable Semiclassical Spectral Asymptotics
Abstract
For an operator \(A:=A_h= A^0(hD) + V(x, hD)\) with a “potential” V decaying as \(|x|\rightarrow \infty \) we establish under certain assumptions the complete and differentiable with respect to \(\tau \) asymptotics of \(e_h(x, x,\tau )\) where \(e_h(x, y,\tau )\) is the Schwartz kernel of the spectral projector.
Victor Ivrii
Chapter 36. Bethe-Sommerfeld Conjecture in Semiclassical Settings
Abstract
Under certain assumptions (including \(d\ge 2)\) we prove that the spectrum of a scalar operator in \(\mathscr {L}^2({\mathbb {R}}^d)\)
$$\begin{aligned} A_\varepsilon (x, hD)= A^0(hD) + \varepsilon B(x, hD), \end{aligned}$$
covers interval \((\tau -\epsilon ,\tau +\epsilon )\), where \(A^0\) is an elliptic operator and B(x, hD) is a periodic perturbation, \(\varepsilon =O(h^\varkappa )\), \(\varkappa >0\).
Further, we consider generalizations.
Victor Ivrii
Chapter 37. 100 years of Weyl’s Law
Abstract
We discuss the asymptotics of the eigenvalue counting function for partial differential operators and related expressions paying the most attention to the sharp asymptotics. We consider Weyl asymptotics, asymptotics with Weyl principal parts and correction terms and asymptotics with non-Weyl principal parts. Semiclassical microlocal analysis, propagation of singularities and related dynamics play crucial role.
We start from the general theory, then consider Schrödinger and Dirac operators with the strong magnetic field and, finally, applications to the asymptotics of the ground state energy of heavy atoms and molecules with or without a magnetic field.
Victor Ivrii
Backmatter
Metadata
Title
Microlocal Analysis, Sharp Spectral Asymptotics and Applications V
Author
Prof. Victor Ivrii
Copyright Year
2019
Electronic ISBN
978-3-030-30561-1
Print ISBN
978-3-030-30560-4
DOI
https://doi.org/10.1007/978-3-030-30561-1

Premium Partner

    Image Credits