§3.3.1 Sturm-Liouville theorem: orthogonal eigenfunctions

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Presentation transcript:

§3.3.1 Sturm-Liouville theorem: orthogonal eigenfunctions Christopher Crawford PHY 416 2014-10-27

Outline Review of eigenvalue problem Linear function spaces: Sturm-Liouville theorem Review of rectangular BVP in term of vectors / eigenstuff Separation of Cartesian variables: Plane waves: exponentials

Vectors vs. Functions Functions can be added or stretched (pointwise operation) Continuous vs. discrete vector space Components: function value at each point Visualization: graphs, not arrows ` `

Vectors vs. Functions ` `

Sturm-Liouville Theorem Laplacian (self-adjoint) has orthogonal eigenfunctions This is true in any orthogonal coordinate system! Sturm-Liouville operator – eigenvalue problem Theorem: eigenfunctions with different eigenvalues are orthogonal

Rectangular box: eigenfunctions Boundary value problem: Laplace equation

Rectangular box: components Boundary value problem: Boundary conditions 7