Spectral Properties of the Laplace Tidal Wave Equation

This paper discusses the eigenvalue problem associated with the weighted Sturm‐Liouville differential equation given, for με(−1,1), by (I) −{( 1 − μ 2 )( s / τ ) + 1( μ 2 − τ 2 )Y ′ ( μ ) } ′ −s 2τ 2( 1 − μ 2 )( s / τ ) − 1 Y ( μ ) = λ( 1 − μ 2 )( s / τ )Y ( μ )where s and τ are parameters, with s a non‐zero integer, and 0 < τ < 1; λ ε C determines the eigenvalues. This problem can be derived from the original form of the Laplace tidal wave equation (II) −{1 − μ 2μ 2 − τ 2y ′ ( μ ) } ′ + { 1 μ 2 − τ 2{ s τ [μ 2 + τ 2μ 2 − τ 2] +s 21 − μ 2} } y ( μ ) = λ y ( μ ) s also valid for με(−1,1), on using quasi‐derivatives and transformation theory. The important feature of interest in ( I ) and ( II ) is that the leading coefficient (1 − μ 2 ) (s/τ)+1 /(μ 2 −τ 2 ) in ( I ) (or (1 − μ 2 )/(μ 2 −τ 2 ) in ( II )) changes sign over the interval (−1,1) which leads to interesting spectral properties of the associated differential operator; in particular it is shown that the spectrum of this operator is unbounded above and below, and is discrete and simple. Additionally the eigenfunctions of the unique operator generated by ( I ) are shown to be complete in the weighted Hilbert function space L 2 w (−1,1) where W (μ) = (1 − μ 2 ) s/τ (με{− 1,1)). This result implies the completeness, in L 2 (−1,1), of the eigenfunctions (commonly called the Hough functions) of the unitarily equivalent operator generated by equation (II).