In this lecture were going to study sturm s theorem, a tool that helps with both of these problems. The eigenvalues of a sturm liouville problem are all of. 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. Questions tagged sturmliouville ask question the sturmliouville equation is a particular secondorder linear differential equation with boundary conditions that often occurs in the study of linear, separable partial differential equations. We mostly deal with the general 2ndorder ode in selfadjoint form. We will merely list some of the important facts and focus on a few of the properties. 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.
In 1910 hermann weyl published one of the most widely quoted papers in analysis 607. Sturm liouville theory the proof shows that if px 6qx then z b a z2p. Il problema di trovare gli autovalori per cui esiste una soluzione non banale dellequazione di sturm. The trigonometric functions have the property that they are orthogonal, that is. Such an equation is said to be in sturm liouville form. Indefinite sturmliouville problems article pdf available in proceedings of the royal society of edinburgh section a mathematics 303. That is, every holomorphic function f for which there exists a positive number m such that. The eigenvalues of a sturm liouville problem are all of multiplicity one. The sturmliouville eigenvalue problems have been discovered by jacques charles. Besides his own research in analysis, algebra, and number theory, liouville was the founder, and for 39 years the editor, of the in. If not, then just as before we could derive by dividing by yx and the boundary term in would vanish since yb 0, and we would obtain z b a yz0. Equivalently, nonconstant holomorphic functions on have unbounded images the theorem is considerably improved by picards little theorem.