Besides his own research in analysis, algebra, and number theory, liouville was the founder, and for 39 years the editor, of the in. This might seem to be a much stronger result than liouville s theorem, but it is actually an easy corollary. In this lecture were going to study sturms theorem, a tool that helps with both of these problems. The eigenvalues of a sturmliouville problem are all of multiplicity one. Teorema di liouville meccanica hamiltoniana wikipedia. Consider a thin rod of length l, perfectly insulated. The rst will have exact closed form solutions because the boundary condtions are very simple, the second will not have closed form solutions, and will need to be solved graphically andor numerically. Quite recently, however, observations show that the spot appears to be shrinking in size. In sturmliouville theory well always use the weighted l2 inner product hf, gi b a wxfxgxdx where f is the complex conjugate of f. Sturmliouville eigenvalue problems motivation the heat. This equation describes the evolution of phase space distribution function for the conservative hamiltonian system. The differential equation is said to be in sturmliouville form or selfadjoint form. A sturm chain is a nite sequence of polynomials p 0x. This proof in ndimensions is completely analogous, except that we need to carefully define an ndimensional flux density liouville s theorem.
As funcoes, e sao parametros e, no caso dito regular, sao continuas no intervalo fechado. Adkins master of science graduate department of mathematics university of toronto 2014 a basic introduction into sturm liouville theory. If we further assume that the heat source q is proportional to the temperature u, q. In 1910 hermann weyl published one of the most widely quoted papers in analysis 607. This book intends to provide a modern survey of some of the basic properties of sturmliouville theory and to bring the reader to the forefront of research on some aspects of this theory. To prove liouvilles theorem, it is enough to show that the derivative of any entire function vanishes. We will merely list some of the important facts and focus on a few of the properties. Examples of separation of variables leading to sturmliouville eigenvalue problems many partial di erential equations which appear in physics can be solved by separation of variables. Introduction to sturmliouville theory and the theory of. It provides a complete description of the system both at equilibrium and also away from equilibrium. Sturmliouville problems 55 this has nontrivial solution for the pair a, b if and only if. Sturmliouville theory mathematical methods in science. Sturmliouville examples math 322 spring 2014 we will go through two examples of solving sturmliouville problems. If the image of f is not dense, then there is a complex number w and a real number r 0 such that the open disk centered at w with radius r has no element of the image of f.
A beautiful consequence of this is a proof of the fundamental theorem of algebra, that any polynomial is completely factorable over the complex numbers. Dozens of papers are published on sturm liouville problems slp every year. In quantum mechanics, eigenvalues correspond to precisely measured quantities. The trigonometric functions have the property that they are orthogonal, that is. Sturms comparison theorem therefore by rolles theorem, there exists a csuch that x1 liouville theoryorthogonal functions or p 0 x p1x. Chapters 3 through 5 present special applications of the sturm. Advanced classical mechanicsliouvilles theorem wikiversity. Second order linear ordinary di erential equations 5 2. Dozens of papers are published on sturmliouville problems slp every year. In other words, suppose that two particles, a and b, follow the same trajectory, except that particle a leads particle b by a finite time or equivalently, there is a finite distance in xp space between the two particles. Sturmliouville theory has remained an intensely active field of research with many applications in mathematics and mathematical physics. Adkins master of science graduate department of mathematics university of toronto 2014 a basic introduction into sturmliouville theory. The solution v is required to satisfy boundary conditions of the type. Liouville s theorem does not imply that every point along a given path has the same density.
Strictly speaking, an operator doesnt have a uniquely determined adjoint, because the adjoint you. Bonudary conditions are either of robin including neumann or dirichlet type. This paper is devoted to studying a qanalogue of sturmliouville eigenvalue problems. Sturm and liouvilles work on ordinary linear differential.
For most values of the parameter, problem 1 has only the trivial solution. Liouville s theorem applies only to hamiltonian systems. We were able to find the eigenvalues of problems 14 explicitly because in each problem the coefficients in the boundary conditions satisfy \\alpha\beta0\ and \\rho\delta. Liouville equation and liouville theorem the liouville equation is a fundamental equation of statistical mechanics. It explains these properties systematically in terms of the sturm. We mostly deal with the general 2ndorder ode in selfadjoint form.
We formulate a selfadjoint qdifference operator in a. Problem 5 isnt, although some authors use a definition of sturmliouville problem that does include it. All secondorder linear ordinary differential equations can be recast in the form on the lefthand side of by multiplying both sides of the equation by an appropriate integrating factor although the same is not true of secondorder partial differential equations, or if y is a vector. Sturm is also famous for a theorem on the number of real zeros of a polynomial, and in addition, did extensive work in physics and mechanics. A proof of liouville s theorem can be found in section 12. Introduction to sturmliouville theory and the theory of generalized fourier series we start with some introductory examples. Examples of separation of variables leading to sturmliouville eigenvalue problems 2 1. In sturmliouville theory, we say that the multiplicity of an eigenvalue of a sturmliouville problem l. This proof of liouville s theorem in a three dimensional phase space uses the divergence theorem theorem in a fashion familiar to most physics majors. Just as the 183637 papers of sturm and liouville started the study of regular slp, the 1910 paper of weyl initiated the investigation of singular slp.
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