The interaction of waves with obstacles is an everyday phenomenon in science and engineering, arising for example in acoustics, electromagnetism, seismology and hydrodynamics. The mathematical theory and technology needed to understand the phenomenon is known as multiple scattering, and this book is the first devoted to the subject. The author covers a variety of techniques, describing first the single-obstacle methods and then extending them to the multiple-obstacle case. A key ingredient in many of these extensions is an appropriate addition theorem: a coherent, thorough exposition of these theorems is given, and computational and numerical issues around them are explored. The application of these methods to different types of problems is also explained; in particular, sound waves, electromagnetic radiation, waves in solids and water waves. A comprehensive bibliography of some 1400 items rounds off the book, which will be an essential reference on the topic for applied mathematicians, physicists and engineers.

Multiple Scattering: Interaction of Time-Harmonic Waves with N Obstacles

/pdf/lectures-on-the-geometry-of-quantization-4mqq1o1gwe.pdf

Lectures on the geometry of quantization

This work is a step towards a non-perturbative continuum definition of quantum field theory (QFT), beginning with asymptotically free two dimensional non-linear sigma-models, using recent ideas from mathematics and QFT. The ideas from mathematics are resurgence theory, the trans-series framework, and Borel-Ecalle resummation. The ideas from QFT use continuity on ${{\mathbb{R}}^1}\times \mathbb{S}_L^1$
 , i.e., the absence of any phase transition as N → ∞ or rapid-crossovers for finite-N, and the small-L weak coupling limit to render the semi-classical sector well-defined and calculable. We classify semi-classical configurations with actions 1/N (kink-instantons), 2/N (bions and bi-kinks), in units where the 2d instanton action is normalized to one. Perturbation theory possesses the IR-renormalon ambiguity that arises due to non-Borel summability of the large-orders perturbation series (of Gevrey-1 type), for which a microscopic cancellation mechanism was unknown. This divergence must be present because the corresponding expansion is on a singular Stokes ray in the complexified coupling constant plane, and the sum exhibits the Stokes phenomenon crossing the ray. We show that there is also a non-perturbative ambiguity inherent to certain neutral topological molecules (neutral bions and bion-anti-bions) in the semiclassical expansion. We find a set of “confluence equations” that encode the exact cancellation of the two different type of ambiguities. There exists a resurgent behavior in the semi-classical trans-series analysis of the QFT, whereby subleading orders of exponential terms mix in a systematic way, canceling all ambiguities. We show that a new notion of “graded resurgence triangle” is necessary to capture the path integral approach to resurgence, and that graded resurgence underlies a potentially rigorous definition of general QFTs. The mass gap and the Θ angle dependence of vacuum energy are calculated from first principles, and are in accord with large-N and lattice results.

Resurgence and trans-series in Quantum Field Theory: the $\mathbb{C}{{\mathbb{P}}^{N-1 }}$ model

Cet article presente une introduction aux methodes resurgentes en asymptotique semi-classique. Les deux premiers chapitres expliquent comment des notions bien connues comme les phenomenes de Stokes pour les developpements BKW a une dimension s'interpretent dans le cadre de la theorie de la resurgence. La suite de l'article est consacree a une etude compiete de la confluence quadratique, i.e., lorsqu'on se place pres des valeurs des parametres ou deux points tournants simples viennent en coincidence.

/pdf/resurgent-methods-in-semi-classical-asymptotics-4s7k9km96u.pdf

Resurgent methods in semi-classical asymptotics

This work is a step towards a non-perturbative continuum definition of quantum field theory (QFT), beginning with asymptotically free two dimensional non-linear sigma-models, using recent ideas from mathematics and QFT. The ideas from mathematics are resurgence theory, the trans-series framework, and Borel-Ecalle resummation. The ideas from QFT use continuity on R^1 x S^1_L, i.e, the absence of any phase transition as N \to infinity, or rapid-crossovers for finite-N, and the small-L weak coupling limit to render the semi-classical sector well-defined and calculable. We classify semi-classical configurations with actions 1/N (kink-instantons), 2/N (bions and bi-kinks), in units where the 2d instanton action is normalized to one. Perturbation theory possesses the IR-renormalon ambiguity that arises due to non-Borel summability of the large-orders perturbation series (of Gevrey-1 type), for which a microscopic cancellation mechanism was unknown. This divergence must be present because the corresponding expansion is on a singular Stokes ray in the complexified coupling constant plane, and the sum exhibits the Stokes phenomenon crossing the ray. We show that there is also a non-perturbative ambiguity inherent to certain neutral topological molecules (neutral bions and bion-anti-bions) in the semiclassical expansion. We find a set of "confluence equations" that encode the exact cancellation of the two different type of ambiguities. We show that a new notion of "graded resurgence triangle" is necessary to capture the path integral approach to resurgence, and that graded resurgence underlies a potentially rigorous definition of general QFTs. The mass gap and the Theta angle dependence of vacuum energy are calculated from first principles, and are in accord with large-N and lattice results.

Resurgence and Trans-series in Quantum Field Theory: The CP(N-1) Model

This book presents the topological and analytical foundations of the theory of Maslov's canonical operator for finding asymptotic solutions of a wide class of pseudodifferential equations. The topology and geometry of Lagrangian manifolds are studied in detail and the connections between Fourier integral operators and canonical operators established. Applications are proposed for the asymptotic solutions to the Cauchy problem and for the asymptotics of the spectra of non-self-dual operators. The authors set out to make more accessible to a wider readership - of specialists in topology, differential equations and functional analysis, including research students - the ideas of Maslov which were much more difficult and opaque in their original published form (1965). The book has been updated, as compared with the Russian edition, by numerous revisions within the text and the addition of two new chapters on more recent work.

Lagrangian manifolds and the Maslov operator

The state of the art of the problem of the analytical continuation of wave fields is reviewed. The problem is a multi- disciplinary subject which involves radiophysics, acoustics, and opticsontheonehand,andmathematicalphysicsandthetheory of differential equation, on the other. The qualitative aspects of the problem are examined. A computational algorithm for field singularities is given. The relation between the singularities and the 'computational catastrophes' of scattering models is dis- cussed. To illustrate the theoretical material, numerous exam- ples are discussed in great detail.

Singularities of continuation of wave fields

A theory of general boundary-value problems is developed for differential operators with symbols not necessarily satisfying the Atiyah-Bott condition that the corresponding obstruction must vanish. A condition ensuring that these problems possess the Fredholm property is introduced and the corresponding theorems are proved.

On general boundary value problems for elliptic equations

CONTENTSIntroduction §1. Technical preliminaries1. Symplectic and contact geometry2. Integral calculus on 3. Homogeneous and formally homogeneous generalized functions on §2. Fourier transforms of homogeneous functions1. Statement of the problem2. The main lemma3. Structure theorem for the Fourier transform of a homogeneous function4. Commutation formulae and the choice of constants §3. Fourier-Maslov integral operators1. The Maslov canonical operator2. Fourier-Maslov integral operators §4. Examples and applications1. Preliminary remarks2. The discontinuity propagation problem3. The discontinuity metamorphosis problem4. Investigation of Green's function of the Cauchy problem §5. Microlocal classification of pseudodifferential operators1. Microlocal equivalence2. Elliptic operators and operators of principal type3. Operators of subprincipal type §6. Equations of principal and subprincipal types1. Equations of principal type2. Proof of Theorem 13. Equations of subprincipal typeReferences

Contact Geometry and Linear Differential Equations

An operator algebra associated with a smooth embedding is constructed. For elliptic elements of this algebra a finiteness theorem (the Fredholm property) is established, and the index is computed. A connection with Sobolev problems is shown.

Relative elliptic theory and the Sobolev problem

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

V. E. Shatalov

Author Tools

Chat about Author