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: -{d\over dx}\left dx} ight +q(x)y=\lambda w(x)y, \qquad (1)

where the functions ''p(x)'', ''q(x)'', and ''w(x)'' are specified at the outset and in the simplest of cases are continuous on the finite closed interval b often together with specified values (also called Boundary Values ) of ''y'' and ''dy''/''dx'' at ''a'' and ''b''. The function ''w''(''x'') is called the "weight" or "density" function. The value of λ is not specified in the equation; finding the values of λ (generally complex numbers) for which there exist a non-trivial solution of (1) satisfying the boundary conditions is part of the problem called the Sturm-Liouville problem (S-L).

Such values of λ when they exist are called the Eigenvalues of the boundary value problem defined by (1) and the prescribed set of boundary conditions. The corresponding solutions (for such a λ) are the Eigenfunction s of this problem. Under normal assumptions on the coefficient functions ''p''(''x''), ''q''(''x''), and ''w''(''x'') above, they induce a Hermitian Differential Operator in some Function Space defined by Boundary Conditions . 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. This theory is important in applied mathematics, where S-L problems occur very commonly, particularly when dealing with linear Partial Differential Equation s which are Separable .


STURM-LIOUVILLE THEORY


The main tenet of Sturm-Liouville theory states that: In the case of regular ''separated boundary conditions'' of the form

: y(a)\cos \alpha - p(a)y^{\prime}(a)\sin \alpha = 0, \qquad (2)

: y(b)\cos \beta - p(b)y^{\prime}(b)\sin \beta = 0, \qquad (3)

:where \alpha, \beta \in [0, \pi),

  • The eigenvalues \lambda_n of the regular Sturm-Liouville problem (1)-(2)-(3) where (''p''(''x'') is differentiable, ''q''(''x'') and ''w''(''x'') are continuous, ''p''(''x'') > 0 and ''w''(''x'') > 0 over the interval) are real and Well Ordered such that


::\lambda_1 < \lambda_2 < \lambda_3 < \cdots < \lambda_n < \cdots o \infty;

  • Corresponding to each eigenvalue \lambda_n is a unique eigenfunction y_n(x) and y_n(x) has exactly n-1 zeros in (a,b);


  • The eigenfunctions are mutually orthogonal and satisfy the orthogonality relation


:: \int_{a}^{b}y_n(x)y_m(x)w(x)\,dx = 0 , m
e n,

:where w(x) is the weight function.

  • If the set of eigenfunctions satisfy the orthogonality relation


:: \int_{a}^{b}y_n(x)y_m(x)w(x)\,dx = \delta_{mn},

:where \delta_{mn}, is the Kronecker Delta then it is said to form an orthonormal set.

  • The eigenvalues of the Sturm-Liouville problem (1)-(2)-(3) can be characterized by the Rayleigh Quotient