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Author: Johannes Sjöstrand Publisher: Springer ISBN: 3030108198 Category : Mathematics Languages : en Pages : 496
Book Description
The asymptotic distribution of eigenvalues of self-adjoint differential operators in the high-energy limit, or the semi-classical limit, is a classical subject going back to H. Weyl of more than a century ago. In the last decades there has been a renewed interest in non-self-adjoint differential operators which have many subtle properties such as instability under small perturbations. Quite remarkably, when adding small random perturbations to such operators, the eigenvalues tend to distribute according to Weyl's law (quite differently from the distribution for the unperturbed operators in analytic cases). A first result in this direction was obtained by M. Hager in her thesis of 2005. Since then, further general results have been obtained, which are the main subject of the present book. Additional themes from the theory of non-self-adjoint operators are also treated. The methods are very much based on microlocal analysis and especially on pseudodifferential operators. The reader will find a broad field with plenty of open problems.
Author: Johannes Sjöstrand Publisher: Springer ISBN: 3030108198 Category : Mathematics Languages : en Pages : 496
Book Description
The asymptotic distribution of eigenvalues of self-adjoint differential operators in the high-energy limit, or the semi-classical limit, is a classical subject going back to H. Weyl of more than a century ago. In the last decades there has been a renewed interest in non-self-adjoint differential operators which have many subtle properties such as instability under small perturbations. Quite remarkably, when adding small random perturbations to such operators, the eigenvalues tend to distribute according to Weyl's law (quite differently from the distribution for the unperturbed operators in analytic cases). A first result in this direction was obtained by M. Hager in her thesis of 2005. Since then, further general results have been obtained, which are the main subject of the present book. Additional themes from the theory of non-self-adjoint operators are also treated. The methods are very much based on microlocal analysis and especially on pseudodifferential operators. The reader will find a broad field with plenty of open problems.
Author: Mats Andersson Publisher: Birkhäuser ISBN: 3319524712 Category : Mathematics Languages : en Pages : 464
Book Description
This book is dedicated to the memory of Mikael Passare, an outstanding Swedish mathematician who devoted his life to developing the theory of analytic functions in several complex variables and exploring geometric ideas first-hand. It includes several papers describing Mikael’s life as well as his contributions to mathematics, written by friends of Mikael’s who share his attitude and passion for science. A major section of the book presents original research articles that further develop Mikael’s ideas and which were written by his former students and co-authors. All these mathematicians work at the interface of analysis and geometry, and Mikael’s impact on their research cannot be underestimated. Most of the contributors were invited speakers at the conference organized at Stockholm University in his honor. This book is an attempt to express our gratitude towards this great mathematician, who left us full of energy and new creative mathematical ideas.
Author: Paula Cerejeiras Publisher: Springer Nature ISBN: 3030875024 Category : Mathematics Languages : en Pages : 663
Book Description
This volume contains the contributions of the participants of the 12th ISAAC congress which was held at the University of Aveiro, Portugal, from July 29 to August 3, 2019. These contributions originate from the following sessions: Applications of dynamical systems theory in biology, Complex Analysis and Partial Differential Equations, Complex Geometry, Complex Variables and Potential Theory, Constructive Methods in the Theory of Composite and Porous Media, Function Spaces and Applications, Generalized Functions and Applications, Geometric & Regularity Properties of Solutions to Elliptic and Parabolic PDEs, Geometries Defined by Differential Forms, Partial Differential Equations on Curved Spacetimes, Partial Differential Equations with Nonstandard Growth, Quaternionic and Clifford Analysis, Recent Progress in Evolution Equations, Wavelet theory and its Related Topics.
Author: John L. Weir Publisher: ISBN: Category : Nonselfadjoint operators Languages : en Pages : 220
Book Description
The aim of this thesis is to study the spectral properties of non-self-adjoint operators via related self-adjoint operators. We consider two different prob-lems: one in which the spectra of a family of non-self-adjoint operators are identical to those of a family of self-adjoint operators and one in which the growth rates of semigroups generated by non-self-adjoint operators are bounded by the growth rates of semigroups generated by related self-adjoint operators. -- In the first problem, we consider a family of non-self-adjoint operators arising in the study of a problem in fluid mechanics in a paper written by Benilov, O'Brien and Sazonov, who argued from numerical and asymptotic evidence that the spectra of the operators are real. We show that the spectra of the operators are identical to the spectra of a family of self-adjoint operators and consist of infinitely many real eigenvalues which accumulate only at infinity. We make use of this correspondence to study certain other properties of the eigenvalues of the non-self-adjoint operators via the self-adjoint operators. In particular, we consider the asymptotic distribution of the eigenvalues for each fixed operator, and the behaviour of each eigenvalue as a small parameter tends to zero. -- In the second, we study the spectral asymptotics of large skew symmetric perturbations of a wide class of Schrodinger operators, generalizing some of the results obtained by Gallagher, Gallay and Nier for the one-dimensional quantum harmonic oscillator. We obtain bounds on the growth rates of the one-parameter semigroups generated by the perturbed operators in terms of the minima of the spectra of related self-adjoint operators. These self-adjoint operators are perturbations of the original Schrodinger operators by non-negative potentials, and we obtain lower bounds on the spectral minima in terms of the behaviour of the potentials at their zeros.
Author: Victor Ivrii Publisher: Springer Science & Business Media ISBN: 3662124963 Category : Mathematics Languages : en Pages : 736
Book Description
The problem of spectral asymptotics, in particular the problem of the asymptotic dis tribution of eigenvalues, is one of the central problems in the spectral theory of partial differential operators; moreover, it is very important for the general theory of partial differential operators. I started working in this domain in 1979 after R. Seeley found a remainder estimate of the same order as the then hypothetical second term for the Laplacian in domains with boundary, and M. Shubin and B. M. Levitan suggested that I should try to prove Weyl's conjecture. During the past fifteen years I have not left the topic, although I had such intentions in 1985 when the methods I invented seemed to fai! to provide furt her progress and only a couple of not very exciting problems remained to be solved. However, at that time I made the step toward local semiclassical spectral asymptotics and rescaling, and new horizons opened.
Author: Erik P. van den Ban Publisher: Springer Science & Business Media ISBN: 0817682449 Category : Mathematics Languages : en Pages : 401
Book Description
Hans Duistermaat, an influential geometer-analyst, made substantial contributions to the theory of ordinary and partial differential equations, symplectic, differential, and algebraic geometry, minimal surfaces, semisimple Lie groups, mechanics, mathematical physics, and related fields. Written in his honor, the invited and refereed articles in this volume contain important new results as well as surveys in some of these areas, clearly demonstrating the impact of Duistermaat's research and, in addition, exhibiting interrelationships among many of the topics.
Author: Rafael del Río Publisher: American Mathematical Soc. ISBN: 0821832972 Category : Mathematics Languages : en Pages : 264
Book Description
This volume gathers the articles based on a series of lectures from a workshop held at the Institute of Applied Mathematics of the National University of Mexico. The aim of the book is to present to a non-specialized audience the basic tools needed to understand and appreciate new trends of research on Schrodinger operator theory. Topics discussed include various aspects of the spectral theory of differential operators, the theory of self-adjoint operators, finite rank perturbations, spectral properties of random Schrodinger operators, and scattering theory for Schrodinger operators. The material is suitable for graduate students and research mathematicians interested in differential operators, in particular, spectral theory of Schrodinger operators.
Author: Johannes Sjöstrand
Author: Johannes Sjöstrand
Author: 
Author: Mats Andersson
Author: Paula Cerejeiras
Author: John L. Weir
Author: John Killeen
Author: Victor Ivrii
Author: Eduardus Marie de Jager
Author: Erik P. van den Ban
Author: Rafael del Río