Question

Consider the following second order linear operator: 82 with Notice, that if instead of 3 we had 2 there, we would get a Legendre operator (whose eigenfunctions are Legendre polynomials). But nothing can be further from it than the operator above. The eigenvalue/eigenfunction problem, emerged in the analysis of vibrations of a particular quant urn liquid. An eigenvalue λ corresponds to an excitation mode of frequency Ω = V The eigenfunction ψ(r) would give a spatial profile of the deviation of the density from the steady state. (a) Show that ulr] = π/1- is a proper weighting function; (b) (Diffeult ) Show that the operator is self-adjoint, with a weighting function w(z), on the space of functions that act on an interval -1 ss+1 and obey the following boundary condition with B=0 . (c) Show that the functions and constitute properly normalized eigenfunctions of L. Find the corresponding eigenvalues. Show that va and vare orthogonal to each other.

Answer 1

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