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94LECTURE 4Sturm-Liouville Eigenvalue ProblemsPossibly one of the most useful facts in mathematics is that a symmetric matrichas real eigenvalues and a set of eigenvectors that form an orthonormal basis. Thisproperty of symmetric matrices has a wide-ranging generalization to the spectralproperties of self-adjoint operators in a Hilbert space, of which the sturm -Liouvilleordinary differential operators are fundamental equations arise throughout applied mathematics. For example,they describe the vibrational modes of various systems, such as the vibrations ofa string or the energy eigenfunctions of a quantum mechanical oscillator, in whichcase the eigenvalues correspond to the resonant frequencies of vibration or energylevels. It was, in part, the idea that the discrete energy levels observed in atomicsystems could be obtained as the eigenvalues of a differential operator which ledSchr odinger to propose his wave problems arise directly as eigenvalue problems in one spacedimension.

LECTURE 4 Sturm-Liouville Eigenvalue Problems Possibly one of the most useful facts in mathematics is that a symmetric matric has real eigenvalues and a set of eigenvectors that form an orthonormal basis.

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