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 Sturm-Liouville theory - Wikipedia, the free encyclopedia
The resulting theory of the existence and asymptotic behavior of the eigenvalues, the corresponding qualitative theory of the eigenfunctions and their completeness in a suitable function space became known as Sturm-Liouville theory.
In mathematics and its applications, a classical Sturm-Liouville equation, named after Jacques Charles François Sturm (1803-1855) and Joseph Liouville (1809-1882), is a real second-order linear differential equation of the form
This theory is important in applied mathematics, where S-L problems occur very commonly, particularly when dealing with linear partial differential equations which are separable. /wiki/Sturm-Liouville_theory   (1345 words) Mathematical formulation of quantum mechanics Article
Prior to the emergence of quantum mechanics as a separate theory, the mathematics used in physics consisted mainly of differential geometry and partial differential equations and to a lesser extent, probability theory.
During the first 10 to 15 years after the emergence of quantum theory (up to about 1925) physicists continued to think of quantum theory within the confines of what is now called classical physics, and in particular within the same mathematical structures.
Fortunately, there is a general mathematical theory of such irreversible operations (see quantum operation) and various physical interpretations of the mathematics. /mathematical_formulation_of_quantum_mechanics.html   (1145 words)

 Search Results for Liouville
Sturm and Liouville examined general linear second order differential equations and examined properties of their eigenvalues, the behaviour of the eigenfunctions and the series expansion of arbitrary functions in terms of these eigenfunctions.
J Lutzen, Joseph Liouville's Contribution to the Theory of Integral Equations, Historia Mathematica 9 (1982), 371-391.
Liouville investigated criteria for integrals of algebraic functions to be algebraic during the period 1832-33. /history/Search/historysearch.cgi?SUGGESTION=Liouville&CONTEXT=1   (3232 words)

 Talk:Sturm-Liouville theory - Wikipedia, the free encyclopedia
The introduction to the theory presented here is of little meaning unless boundary conditions are introduced and this should be done at the outset, in the preamble.
I think it's a good idea to have a multiple variable example of a S-L problem, and the wave equation is fine with me, but I think the wave operator (what do physicists call it?) and the S-L connection should be made explicit.
I've just rewritten the opening paragraph here to conform with existing definitions and terminology. /wiki/Talk:Sturm-Liouville_theory   (392 words)

Geometry, spectral theory, groups, and dynamics : proceedings in memory of Robert Brooks, December 29, 2003-January 2, 2004, January 5-9, 2004, Technion-Israel Institute of Technology, Haifa, Israel / Michael Entov, Yehuda Pinchover, Michah Sageev, editors.
Recent advances in operator theory and its applications : the Israel Gohberg anniversary volume : International Workshop on Operator Theory and its Applications, IWOTA 2003, Cagliari, Italy / Marinus A. Kaashoek, Sebastiano Seatzu, Cornelis van der Mee, editors.
Global theory of minimal surfaces : proceedings of the Clay Mathematics Institute 2001 Summer School, Mathematical Sciences Research Institute, Berkeley, California, June 25-July 27, 2001 / David Hoffman, editor. /newbook/800/new.html   (2881 words)

 History of Operator Theory
The earliest significant appearance of eigenvalues in connection with differential equations was in the theory developed by Charles François Sturm in 1836 and Joseph Liouville in 1838.
Sturm-Liouville theory was the beginning of what we now refer to as the spectral theory of ordinary differential operators.
In the first textbook on operator theory, Théorie des Opérations Linéaires, published in Warsaw 1932, Stefan Banach states that the subject of the book is the study of functions on spaces of infinite dimension, especially those he coyly refers to as spaces of type B, otherwise Banach spaces (definition). /opthy/OpHistory.html   (2635 words)

 Syllabus -- Applied Mathematics 2, Spring 2000
The Sturm-Liouville theory includes all the special cases of eigenfunctions/eigenvalues that we studied last term, and explains, in a sense, why it is always possible to find a collection of eigenfunctions satisfying orthogonality relations for suitable boundary value problems.
We will begin with a general theory of eigenfunctions and eigenvalues of boundary value problems for 2nd order ODE, developed first by the mathematicians Sturm and Liouville in the 19th century.
Next, we will consider two applications of the Sturm-Liouville theory -- the circular drumhead, where the normal modes of vibration involve a new family of eigenfunctions you have not seen before (called the Bessel functions) and non-homogenous problems where eigenfunction expansions are useful to find solutions and understand their properties. /~little/Applied9900/AM200Syl.html   (744 words)

 MTH-3A36 : Transform Theory
Sturm and Liouville made some progress in demonstrating the completeness of such sequences, and ad hoc proofs for particular cases can be constructed.
The broadly unifying theme of the material covered is the theory of integral transforms such as Laplace and Fourier transforms.
Such techniques are particularly suitable for the inversion integrals arising from transform theory. /maths/syllabuses/9900/3A3600.html   (526 words)

 AE Syllabus for MAE 105
Fourier series, Sturm Liouville theory, elementary partial differential equations, integral transforms with applications to problems in vibration, wave motion, and heat conduction. /aerospace/AE_syllabi/MAE_105.html   (452 words)

 QM and Sturm Liouville Theory - Physics Help and Math Help - Physics Forums
Sturm and Liouville in the nineteenth century introduced the technology of linear algebra into the theory of differential equations and showed the importance of eigenvalues to the solutions.
i was curious if understanding of this Sturm and Liouville theory is pertinent to the understanding of QM and what essence would one miss if one does not know this this...
in Sturm and Liouville theory is also very important. /showthread.php?t=87078   (239 words)

Thus, in an interesting connection with operator theory, degenerate Sturm-Liouville operators give rise to self-adjoint operators on a Kre\v{i}n space whose spectrum is all of $\mathbb{C}$ (see \cite{alm}, Section 4] for more details about this connection.
Published November 12, 2004.} \thanks{Partially supported by a Research Grant from NSERC Canada} \subjclass[2000]{34B24, 34L05} \keywords{Sturm-Liouville theory; eigenvalues; degenerate operators; \hfill\break\indent spectral theory; Dirichlet problem} \begin{abstract} Consider the Dirichlet eigenvalue problem associated with the real two-term weighted Sturm-Liouville equation $-(p(x)y')' = \lambda r(x)y$ on the finite interval $[a,b]$.
However, {\it there is no Sturm oscillation theorem} (or even its extension by Haupt and Richardson, cf., \cite{abm}) for the eigenfunctions as we show presently. /EMIS/journals/EJDE/Monographs/Monographs/Volumes/2004/130/mingarelli-tex   (2573 words)

In this paper, we use the Sturm-Liouville theory to compute Green functions within a rigorous mathematical theory.
The Sturm-Liouville theory deals directly with complex energies in a way that keep track of the ``in'' and ``out'' boundary conditions.
We shall show that both the Sturm-Liouville theory and the formal treatment yield the same Green functions. /mp_arc/papers/02-40   (1265 words)

Subject: Sturm Colloquium From the web site: Colloquium on the occasion of the 200th anniversary of Charles François Sturm and Workshop on Sturm-Liouville theory Geneva, Switzerland September 15-19, 2003 Geneva wishes to honour the memory of Charles Sturm on the occasion of the 200th anniversary of his birth.
In the second part of the colloquium a few invited specialists on Sturm-Liouville theory will discuss the developments in this branch of mathematics during the 20th century, as well as their applications in the natural sciences, in particular in physics.
In the first one historians of science will analyze the scientific work of Charles Sturm, whereas the second one will be devoted to modern and contemporary aspects of Sturm-Liouville theory. /opsf/siamopsfnet102   (1311 words)

- Sturm-Liouville theory shows that there's a more general class of eigenvalue ODE problems whose solutions can be used, just like before, to build infinite sums which will approximate "any" other function f(x) that satisfies the boundary conditions that correspond to that particular problem.
- Applications of Sturm-Liouville theory are exactly the same as for Fourier theory: in particular, we use it to find the coefficients needed in our "step 3" (see above) for solving the heat equation via the separation of variables method.
In theory however it is always possible to solve it, in exactly the same way as usual, using the usual orthogonality argument. /~pacini/4581/4581_syllabus.html   (2531 words)

Boundary value problems for heat and wave equations, separation of variables, eigenfunction expansions, Sturm-Liouville theory and Fourier series, method of characteristics, Laplace equations. /math_stat/Syllabi_MathDept/Math531PartialDiffEq.doc   (705 words)

 MIT OpenCourseWare Mathematics 18.152 Introduction to Partial Differential Equations, Fall 2004 Home
It also covers the Sturm-Liouville theory and eigenfunction expansions, as well as the Dirichlet problem for Laplace's operator and potential theory.
Your use of the MIT OpenCourseWare site and course materials is subject to the conditions and terms of use in our Legal Notices section.
This course analyzes initial and boundary value problems for ordinary differential equations and the wave and heat equation in one space dimension. /OcwWeb/Mathematics/18-152Fall-2004/CourseHome   (118 words)

 Oscar Bruno - ACM101c
Sturm-Liouville theory, Self-Adjoint operators and Green's functions examples by Dave.
Hilbert spaces, operator theory, spectral theory, Sturm-Liouville problem.
Greens Functions and Transform pairs examples and theory by Dave. /~acm101/2005/SPRING   (284 words)

 The Classical Free-Reed, Inc. -- Alternative Tuning Systems and Free-Reeds
I was first introduced to the concept of "vibrational nonuniformity" in the context of a nonuniform string being used as a motivating example in Sturm-Liouville Theory that, while not in every textbook, it does appear in a good number on the subject.
Sadly though it seems few physics departments ever mention acoustics and the applications students will make of Sturm-Liouville theory will usually be in quantum mechanics and electro dynamics since acoustics is only occasionally taught in physics departments, or mechanical engineering and sometimes in electrical engineering...
This would be done with hopes of finding "desirable" nonharmonic timbres of which scales and tuning systems could then be based on. /free-reed/essays/ats_free-reed.html   (640 words)

 AMCA: Singular Sturm-Liouville theory on manifolds by Robert McOwen
AMCA: Singular Sturm-Liouville theory on manifolds by Robert McOwen
The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts. /c/a/h/s/05.htm   (126 words)

 Sturm bicentenary book
Charles Sturm and the development of Sturm-Liouville theory in the years 1900 to 1950 (434kb)
Spectral Theory of Sturm-Liouville Operators on Infinite Intervals: A Review of Recent Developments (392kb)
Sturm's 1836 Oscillation Results: Evolution of the Theory (453kb) /sturmbook/list.html   (106 words)

 Physics 452 Lecture Schedule
While the first two topics will follow the textbook (Arfken and Weber), the group theory material will go beyond what is covered in Ch.
I will provide lecture notes for the group theory lectures.
However, it is still important that you attend the lectures. /p452/syllabus452.html   (148 words)

 Search for Sturm books:
Sturm Und Drang: The Robbers, the Soldiers, Storm and Stress and the Child Murderess (German Library)
Sturm auf Moskau: Von Finnland bis zum Schwarzen Meer (Assault on Moscow: From Finland to the Black Sea)
Protestant Politics: Jacob Sturm (1489-1553) and the German Reformation (Studies in German Histories) /books/search/2-Sturm.html   (177 words)

 Advanced Engineering math 3350
This includes perturbation theory for polynomial and transcendental equations including the method of laplace inversion.
This includes: Poincare-Lindstedt, averaging, multiple-scales and boundary layer theory.
This chapter includes a discussion of variational problems that are solved by obtaining an appropriate Euler equation or equations. /~gilliam/classes_s01/m5311_s01.htm   (558 words)

 MAST-667 Spring 1999
The formal derivation of the wave equation as a generalized Sturm-Liouville theory
In this course we will cover the general wave mechanics theories and their applications in water waves, acoustic/seismic waves, and electromagnetic waves.
WAVE MECHANICS (MAST-693) is a required course for POSE program. /~badiey/mast693.html   (191 words)

In conjunction with solving boundary value problems, we will introduce the Fourier series and the Sturm-Liouville theory.
The separated solutions to the equations of heat conduction and wave propagation are derived and discussed in detail.
Brief review of MATH 238 topics (3 meetings). /syllabi/fall/343syl.htm   (405 words)

Sturm-Liouville Theory A postscript file summarizing a few issues in Sturm-Liouville theory and associated boundary value problems.
2nd order Linear to SL code Some code to convert non-self adjoint operators to self adjoint Sturm Liouville operators.
Regular SL eigenvalue problem Some code to solve regular Sturm-Liouville eigenvalue problems with separated boundary conditions. /~pernarow/M560/2000/M560.html   (511 words)

 Inverse Eigenvalue Problems
The classic problem in inverse Sturm-Liouville theory is to determine (one of) the coefficients p, q or r in the second order equation
We must impose boundary conditions, and we assume these have the form /~william.rundell/research/isl/invspectral.html   (707 words)

 Northeastern State University
COURSE PURPOSE: The purpose of this course is to introduce the advanced undergraduate mathematics and/or physics major to the theory and applications of classical elementary partial differential equations.
This course will provide the necessary background, expertise, and experience that is required to solve elementary classical partial differential equations in 2 and 3 dimensions.
CATALOG DESCRIPTION OF COURSE: Series solution of ordinary equations, Fourier series, classical second order partial differential equations, heat equation, wave equation, and Laplace's equation. /~math/courses/nca4123.htm   (600 words)

Lab 6: Sturm-Liouville Theory, Eigenfunction Expansions, and Special Functions
There are a special class of differential equations used throughout engineering and physics that have the form /~keady/Matlab/ONlabs/l06sym.html   (911 words)

 mathematical physics
Sturm-Liouville Theory, Past and Present, Birkhäuser, Basel, 2005.
Sturm-Liouville Theory Past and Present, Birkhäuser, Basel, 2005.
del Río, C. Villegas-Blas (Eds.), Spectral Theory of Schrödinger Operators, /~hinz/mathphys.html   (307 words)

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