Scattering Operator, Eisenstein Series, Inner Product Formula and "Maass-Selberg" Relations (Memoirs of the American Mathematical Society)

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The Dirac Equation is a modern presentation of the relativistic quantum mechanics of electrons intended to make new mathematical results accessible to a wider audience. It treats in some depth the relativistic invariance of a quantum theory, self-adjointness and spectral theory, qualitative features of relativistic bound and scattering states, and the external field problem in quantum electrodynamics, without neglecting the interpretational difficulties and limitations of the theory. A novel feature is the extensive use of supersymmetric quantum mechanics in the discussions of, e.g. the nonrelativistic limit, Fredholm and Witten indices, the relativistic Coulomb problem, particles with anomalous moments, behaviour in the presence of magnetic fields, and soliton theory. The book is aimed at scientists and graduate students interested in quantum mechanics who want to learn about the mathematical foundations, the physical interpretation and important applications of Dirac's electron theory.

An Introduction to Inverse Scattering and Inverse Spectral Problems (Monographs on Mathematical Modeling and Computation)

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An Introduction to Inverse Scattering and Inverse Spectral Problems (Monographs on Mathematical Modeling and Computation)
Khosrow Chadan , David Colton , Lassi Päivärinta , and William Rundell
Manufacturer: Society for Industrial Mathematics
ProductGroup: Book
Binding: Paperback

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ASIN: 0898713870

Determining Spectra in Quantum Theory (Progress in Mathematical Physics)

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Determining Spectra in Quantum Theory (Progress in Mathematical Physics)
Michael Demuth , and Maddaly Krishna
Manufacturer: Birkhäuser Boston
ProductGroup: Book
Binding: Hardcover

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ASIN: 0817643664

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The spectral theory of Schrödinger operators, in particular those with random potentials, continues to be a very active field of research. This work focuses on various known criteria in the spectral theory of selfadjoint operators in order to identify the spectrum and its components à la Lebesgue decomposition.

Key features and topics:

* Well-developed exposition of criteria that are especially useful in determining the spectra of deterministic and random Schrödinger operators occurring in quantum theory

* Systematically uses measures and their transforms (Fourier, Borel, wavelet) to present a unifying theme

* Establishes criteria for identifying the spectrum

* Examines a series of applications to show point spectrum and continuous spectrum in some models of random operators

* Presents a series of spectral-theoretic results for the perturbed operators introduced in the earlier chapters with examples of localization and delocalization in the theory of disordered systems

* Presents modern criteria (using wavelet transform, eigenfunction decay) that could be used to do spectral theory

* Unique work in book form combining the presentation of the deterministic and random cases, which will serve as a platform for further research activities

This concise unified presentation is aimed at graduate students and researchers working in the spectral theory of Schrödinger operators with either fixed or random potentials in particular. However, given the large gap that this book fills in the literature, it will serve a wider audience of mathematical physicists in its contribution to works in spectral theory.

Differential Operators and Spectral Theory: M. Sh. Birman's 70th Anniversary Collection (American Mathematical Society Translations Series 2)

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Differential Operators and Spectral Theory: M. Sh. Birman's 70th Anniversary Collection (American Mathematical Society Translations Series 2)

Manufacturer: American Mathematical Society
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Binding: Hardcover

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This volume contains a collection of original papers in mathematical physics, spectral theory, and differential equations. The papers are dedicated to the outstanding mathematician, Professor M. Sh. Birman, on the occasion of his 70th birthday. Contributing authors are leading specialists and close professional colleagues of Birman.

The main topics discussed are spectral and scattering theory of differential operators, trace formulas, and boundary value problems for PDEs. Several papers are devoted to the magnetic Schrödinger operator, which is within Birman's current scope of interests and recently has been studied extensively. Included is a detailed survey of his mathematical work and an updated list of his publications.

This book is aimed at graduate students and specialists in the above-mentioned branches of mathematics and theoretical physicists. The biographical section will be of interest to readers concerned with the scientific activities of Birman and the history of those branches of analysis and spectral theory where his contributions were important and often decisive.

Features:

The first detailed survey of Birman's mathematical work; includes an updated bibliography.

New material on the history of some branches of analysis.

Prominent authors: Lieb, Agmon, Deift, Simon, Ladyzhenskaya, and others.

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"This useful volume, based on the Taniguchi International Workshop held recently in Sanda, Hyogo, Japan, discusses current problems and offers the mostup-to-date methods for research in spectral and scattering theory."

Spectral and Scattering Theory and Applications (Advanced Studies in Pure Mathematics)

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Spectral and Scattering Theory and Applications (Advanced Studies in Pure Mathematics)

Manufacturer: Amer Mathematical Society
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Binding: Hardcover

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Book Description

This book offers a new interpretation of Plato's early and middle dialogues as the expression of a unified philosophical vision. Whereas the traditional view sees the dialogues as marking successive stages in Plato's philosophical development, we may more legitimately read them as reflecting an artistic plan for the gradual, indirect and partial exposition of Platonic philosophy. The magnificent literary achievement of the dialogues can be fully appreciated only from the viewpoint of a unitarian reading of the philosophical content.

Customer Reviews:

What do I know? I think this book is original........2002-03-04

... I'm not finished with Kahn's book, but I find the central thesis fascinating. I had not considered it before. Almost everyone seems to believe that Plato's philosophy developed over time. It seems possible to construct a time line of dialogues with the "early" ones representing Socrates more than the later ones.

Who before Kahn has ever suggested an "ingressive" approach, where Plato's philosphy is fully-formed, but only revealed in pieces? Perhaps others, I do not know. But I think the model Kahn suggests opens up a whole line of thinking about Plato. So Plato didn't discuss "recollection" in the Meno without having the fully fledged idea of Forms in mind. I've always had the impression that scholars were saying that Plato's doctrine of "recollection" was the most advanced position he had at that time, as if "Forms" hadn't occured to him yet.

Anyway, I like where Kahn is going. He may not be expressing the "common opinion," but he is correct to tie the literary qualities in with the philosophy. ... I could be wrong.

"...I'd always thought of you as quick.".......2001-07-02

I had written a lengthy review of this work but, since it has never appeared on this site, I will attempt to reprise it in a more condensed version. Charles Kahn is a highly respected scholar of ancient philosophy at the University of Pennsylvania. Unfortunately, this book does him no credit and actually calls into question his very ability to read Plato. Put as broadly as possible, Kahn sees the platonic corpus as an attempt by Plato to gradually educate his readers in his doctrines. The progress he traces is from the "earliest" works (eg. the Apology) to the Republic, which he regards as the fullest revelation of Plato's teachings. To that end he proceeds to ransack every dialogue within this period for indications of a "developmental" approach by Plato. Differences between dialogues are papered over or fobbed off on the vague assertion that Plato wasn't engaged in a coherent discussion of the topic-in-question at this point in his writings because he didn't believe that his readership could understand it. Essentially Kahn has inverted the usual "developmental" approach to Plato (ala Gregory Vlastos) by assuming that Plato's writings evolved while his philosophy did not. While this assumption is preferable it does not aid Kahn in his interpretation.

This book is a classic example of a scholar letting his critical apparatus (and prejudices) get in the way of the necessary task of sustained, careful exegesis. Kahn has absolutely no "feel" for the "literary" elements of the dialogues and he cannot give any reasons for the fact that Socrates's discussions are different when he is speaking with different interlocutors. Kahn ignores very important details for the sake of his pet thesis and it is invariably those details which disprove Kahn's readings. Why does Diotima lecture Socrates about eros in the "Symposium" but it is Socrates who does the lecturing in the "Phaedrus?" Kahn is incapable of asking or answering this question. Plato was a writer of considerable comedic talents but Kahn pays little or no attention to this. He is also enamored of making embarassing statements about Plato being a "mystic" and a "metaphysician" who is not interested in the everyday world. As Kahn never defines what a "mystic" would be it is very difficult to know whether he is referring to Madame Blavatsky or Plotinus. In addition, Plato's consistent engagement with politics and its relationship to philosophy disproves such assertions. Furthermore, Kahn's dismissal of Xenophon as "unphilosophical" raises the question of whether the ivy-league professor is being careless or just incompetent. Recent work on Xenophon has revealed a thinker of subtle complexity who was well regarded by men such as Cicero, Machiavelli, and Sir Philip Sidney. Kahn's inability to understand Xenophon is one in a series of grave flaws which capsize this work. Put as baldly as possible, this is a bad book--perhaps the worst I have read in several years--which should never have been published. The hypothesis is absurd and the analysis very shoddy. What other readers seem to interpret as "boldness" is really just zealous belief in a questionable interpretation (ie. monomania). Plato is far too subtle a thinker and writer for Kahn to grasp so the professor decided to construct an effigy of Plato which he then sets alight, believing that it is the real thing. Avoid this work at all costs and, instead, spend your money on Sayre's "Plato's Literary Garden" or Sallis's "Being and Logos" where you will be treated to a wonderfully complex discussion about the ancient sage and his writings.

Controversial and Challenging.......2000-11-19

This is one of the best books on Plato that I have read. Kahn's thesis--that Plato's early and middle dialogues present a unified philosophical vision that is gradually revealed from dialogue to dialogue (what Kahn calls the "ingressive method")--is a new twist on the unitarian thesis that the Platonic corpus gives no evidence for the sort of philosophical development that has been spotted by interpreters such as Grote, Campbell, Vlastos, Owen, and many others (probably most others, in fact). But even if one is a developmentalist at heart, one can benefit greatly from reading this book. The approach is both philosophical and scholarly, of use both to the philosopher and to the classicist. Even when it is difficult to agree with Kahn (for example, he holds that the Gorgias is an earlier work than the Protagoras, in spite of what appears to be a more complex moral psychology and a more sensitive treatment of the hedonist thesis in the former), grappling with his arguments can be both a challenge and a thrill. Rarely does disagreement serve to educate so well.

It is disappointing, though fully understandable, that this book does not treat the late dialogues. There are hints here and there that Kahn thinks he could extend his thesis further, but his treatment of the Pheadrus in the last chapter is more promisory than productive.

Hubris to the max.......2000-01-11

When the first empirical experiments confirmed Einstein's theory of relativity, story has it he was unphased. "If it had been otherwise" he is quoted as saying, "I'd feel sorry for God." If Kahn's interpretation of Plato is not correct, I feel sorry for Plato. To hear Kahn tell it, Plato is a great genius who did not, as modern scholarly orthodoxy holds, develope his point of view over time, but rather developes his reader over time to accept his ideas. Kahn believes that the decisive influence on Plato's life (other than, of course, Socrates) was the coup of the Thirty. These aristocrats overturned the Athenian democracy and, instead of ruling nobly, showed Plato how depraved and stupid obligarchy can be. The worst of these revolutionaries were Plato's brother and cousin. The only is bright spot was Socrates' brave stand against the tyrants only to get it in the neck once democracy was restored. The problem was that the ancient Greek religion was an aristocratic, heroic religion (see Homer) which encouraged the aristocrats to behave like barbarians. What Plato was trying to do was introduce a new religion (philosophy) which would civilize the aristocracy. To prove his point, Kahn must progress from Ion and Hippias Minor, through Gorgias (where he states his question fully) to the aporetic dialogues (Lysis, Meno, Charmnides) to the Protagoras (the most problematic of the dialogues to fit into his theory)and finally to goal the great middle dialogues (Symposium, Phaedo and Republic). It's quite a ride.but if you can hold on, well worth it.

Plato's single literary project.......1999-12-27

I do not believe I can capture the sheer audacity and interpretive hubris of this book so I quote the author: "The anonymity of the dialogue form, together with Plato's problematic irony in the presentation of Socrates, makes it impossible for us to see through these dramatic works in such a way as to read the mind of their author. To suppose that one can treat these dialogues as a direct statement of the author's opinion is what I call the fallacy of transparency, the failure to take account of the doctrinal opacity of these literary texts. What we can and must attempt to discern, however, is the artistic intention with which they were composed. For in this sense the intention of the author is inscribed in the text. It is precisely this intention that my exegesis is designed to capture, by construing the seven threshold dialogues together with Symposium and Phaedo as a single complex literary enterprise culminating with the Republic. And that means to see this whole group of dialogues as the multi-faceted expression of a single philosophical view." (page 42)

Most scholars understand Plato's dialogues in terms of philosophical stages, that is to say, Plato had an early period, when his thought was dominated by Socrates, later came the middle period, culminating in the Republic, when he came, more and more to express his own ideas, and finally a period where he turns against Socrates entirely. But Kahn wants to know what if Plato had the plan of the dialogues mapped out in advance. What if he was critical of Socrates from the beginning? What if Socrates is not his spokesman, but an object of his criticism? Certainly, if Kahn's interpretation stands up, he has Occam's razor on his side. If Plato's dialogues break down into three groups on stylistic grounds, does that justify the assumption on that basis, the three groups date from differing periods, when Plato held differing points of view? Or are we better served to believe that these grouping constitute a literary device intentionally employed by Plato to advance a single, unchanging program? Moreover, how do we know Plato preserves the historical Socrates in his writing, or is even interested in doing so? My training is in biblical studies so I am glad to hear someone asking the same questions about the dialogues of Plato that have been commonplaces in relation to the Gospels for over a century? Are we to assume that because Socrates never performed miracles that that justifies shoddy scholarship? Certainly, from the perspective of Biblical scholars, who have been disavows the biographical nature of the Gospels for decades, reading books like Gregory Vlatos Socrates, Ironist and Moral philosopher gives one the impression that it was written in another century. And it was written (or at least published) in this decade.

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This digital document is an article from The Review of Metaphysics, published by Philosophy Education Society, Inc. on September 1, 1999. The length of the article is 834 words. The page length shown above is based on a typical 300-word page. The article is delivered in HTML format and is available in your Amazon.com Digital Locker immediately after purchase. You can view it with any web browser.

Citation Details
Title: Plato and the Socratic Dialogue: The Philosophical Use of a Literary Form.(Review)
Author: Thomas A. Blackson
Publication: The Review of Metaphysics (Refereed)
Date: September 1, 1999
Publisher: Philosophy Education Society, Inc.
Volume: 53 Issue: 1 Page: 172

Article Type: Book Review

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