in mathematics, a certain class of partial differential equations (partial differential equation) (PDEs) subject to extra constraints, known as boundary values (boundary value), on the solutions. Such equations are common in both classical physics (e.g., thermal conduction) and quantum mechanics (e.g., Schrödinger equation) to describe processes where some external value (boundary value) is held constant while the system of interest transmits some form of energy.

In the mid-1830s the French mathematicians Charles-François Sturm (Sturm, Charles-François) and Joseph Liouville (Liouville, Joseph) independently worked on the problem of heat conduction through a metal bar, in the process developing techniques for solving a large class of PDEs, the simplest of which take the form 【p(x)y′】′+【q(x)−λr(x)】y=0 where y is some physical quantity (or the quantum mechanical wave function) and λ is a parameter, or eigenvalue, that constrains the equation so that y satisfies the boundary values at the endpoints of the interval over which the variable x ranges. If the functions (function) p, q, and r satisfy suitable conditions, the equation will have a family of solutions, called eigenfunctions, corresponding to the eigenvalue solutions.

For the more-complicated nonhomogeneous case in which the right side of the above equation is a function, f(x), rather than zero, the eigenvalues of the corresponding homogeneous equation can be compared with the eigenvalues of the original equation. If these values are different, the problem will have a unique solution. On the other hand, if one of these eigenvalues matches, the problem will have either no solution or a whole family of solutions, depending on the properties of the function f(x).