### Abstract

The eigenvalues and eigenfunctions of self‐adjoint Sturm–Liouville problems with a simple pole on the *i* *n* *t* *e* *r* *i* *o* *r* of the interval [*A*, *B*] are investigated. Three general theorems are proved and it is shown that as *n*→∞, the eigenfunctions more and more closely resemble those of an ordinary Sturm–Liouville problem and λ_{ n } ∼−*m* ^{2}π^{2}/(*B*−*A*)^{2}, just as if there were no singularity. The low‐order modes, however, differ drastically from those of a nonsingular eigenproblem in that (i) both eigenvalues and eigenfunctions are *c* *o* *m* *p* *l* *e* *x* (despite the fact the problem is self‐adjoint), (ii) the real and imaginary parts of the *n*th eigenfunction may both have ever‐increasing numbers of interior zeros as *B*→∞, instead of just (*n*−1) zeros, and (iii) as *B*→∞, the eigenvalues for all small *n* may cluster about a common value in contrast to the widely separated eigenvalues of the corresponding nonsingular problem. These results are general, but in order to present quantitative solutions for the low‐order modes, too, special attention is given to the particular case *u* ^{″}+(1/*x*−λ)*u* = 0, (1) with *u*(*A*) = *u*(*B*) = 0 where λ is the eigenvalue and *A* and *B* are of *o* *p* *p* *o* *s* *i* *t* *e* signs. For small *n*, one can obtain the approximation λ_{ n }∼exp[(1+3^{1/2} *i*)*d* _{ n }/(2*B* ^{1/3})]/*B*, (2) where *d* _{ n } is the *n*th root of the Airy function Ai(−*z*). The imaginary part of (2) shows explicitly how profoundly the interior pole has modified the structure of the eigenproblem. The WKB method, which was used to derive (2), is shown to be accurate for all *n*. The WKB analysis is of some interest in and of itself. Although the number of WKB ’’transition’’ points is the same as for the half‐century old quantum harmonic oscillator (two), the substitution of the interior pole for one of the turning points has a profound (and fascinating) impact on both the WKB formalism and the numerical results. Thus, although this problem was motivated by the physics of hydrodynamic waves, it is also an extension to both classical Sturm–Liouville theory and to the WKB treatment of eigenvalue problems.

Commenting has been disabled for this content