Sturm–Liouville theory

A classical Sturm- Liouville problem (after Joseph Liouville and Charles -François Sturm) is the following eigenvalue problem from calculus: Find all complex numbers for which the differential equation

On the interval possesses a solution, the boundary conditions of the

Sufficient.

Performs to the linear operator of the form

A, the Sturm-Liouville operator, the eigenvalue equation can be made using methods from functional analysis ( spectral ) in the Hilbert space of square integrable functions with respect to the weight functions are treated.

If the interval is compact and the coefficients are integrable, then one speaks of a regular Sturm- Liouville problem. If the interval is unbounded or the coefficients only locally integrable, then one speaks of a singular Sturm- Liouville problem.

Regular Sturm- Liouville problems

The eigenvalue equation

With integrable real functions, together with boundary conditions of the form

Is called a regular Sturm- Liouville problem on the interval, if this interval is finite.

In case one speaks of Dirichlet boundary conditions and in the case of Neumann boundary conditions.

For the regular Sturm- Liouville problem is that there is a countable sequence of real eigenvalues ​​, which diverges to:

The eigenvalues ​​behave asymptotically as ( Weyl asymptotics ):

The corresponding eigenfunctions form an orthonormal basis in the Hilbert space of square integrable functions with respect to the weight functions.

Oszillationssatz

The Sturm'sche Oszillationssatz states that in the case of Dirichlet boundary conditions has exactly the th eigenfunction zeros in the interval. In particular, the lowest eigenfunction can be chosen positive.

Example

A simple example is the differential equation

On the interval, along with the Dirichlet boundary conditions

The eigenvalues ​​are

And the normalized eigenfunctions are

The corresponding eigenfunction development is the Fourier series.

Mathematical Theory

The appropriate mathematical framework is the Hilbert space with the scalar product

In this room is a self-adjoint operator if he is on the set of ( in the sense of weak derivative) differentiable functions that satisfy the boundary conditions, defined as:

Here denotes the set of absolutely continuous functions on. Since an operator is unlimited, we consider the resolvent

Which can not be an eigenvalue. It turns out that the resolvent is an integral operator with continuous kernel ( the Green's function of the boundary value problem). Thus, the resolvent is a compact operator, and the existence of a countable sequence of eigenfunctions follows from the spectral theorem for compact operators.

The relationship between the eigenvalues ​​of the resolvent and follows, as is equivalent to having.

Singular Sturm- Liouville problems

If the above conditions are not met, then one speaks of a singular Sturm- Liouville problem. The spectrum is then generally no longer only of eigenvalues ​​and also has a continuous component. There is also generalized eigenfunctions, and the corresponding eigenfunction development is an integral transformation (see Fourier transform instead of the Fourier series ).

Changes or the sign, then one speaks of a indefinite Sturm-Liouville problem.

Application

• The case corresponds to the one-dimensional time-independent Schrödinger equation.
• The separation of variables for solving partial differential equations leads to Sturm-Liouville problems.