Twin convergence regions for continued fractions

V. F. Cowling, Walter Leighton, W. J. Thron
1944 Bulletin of the American Mathematical Society  
1. In the theory of hydrodynamic stability, eigenvalue problems of the form 1 (1.1) Lu -\-Mu = \u X arise [l, p. 430 ]. Here, L and M denote ordinary differential operators, the order of L exceeds that of M, and the boundary conditions are such that L is self-adjoint. One of the questions of interest is whether there exist eigenvalues of this problem and, if so, whether the corresponding eigenfunctions are complete. Replacing X by 1/X, it is easy to see that if L~l exists, (1.1) is equivalent
more » ... .1) is equivalent to where A =L_1 and B= -L~lM are compact, and A is symmetric. In this note, we shall consider the question of the completeness of the eigenfunctions of the following generalization of (1.2) : (1.3) Xw = Au + \aBxu, where a> 1, A is compact and symmetric, and B\, which, as the notation indicates, may depend on X, is merely bounded. More precise Received by the editors May 11, 1962.
doi:10.1090/s0002-9904-1944-08142-1 fatcat:i6d5wxr7sbdwdlhcavjssi45je