Spectral properties of second order singularly perturbed boundary value problems with turning points

We study degeneration for ¿ + 0 of the two-point boundary value problems t±u := ((au')' + bu' + cu) ± xu' - ¿u = h, u(±1) = A ± B, and convergence of the operators T+ and T- on 2(-1, 1) connected with them, T±u := t±u for all uD(T±, D(T±) := {u L2(-1, 1) u¿ L2(-1, 1) & u(-1) = u(1) = O}, T0+u: = xu' for all uD(TO+), D(TO+) := {u L2(-1, 1) xu' L2(-1, 1) & u(-1) = u(1) = O}. Here is a small positive parameter, ¿ a complex "spectral" parameter; a, b and c are real 8-functions, a(x) ¿ > 0 for all x [-1, 1] and h is a sufficiently smooth complex function. We prove that the limits of the eigenvalues of T+ and of T- are the negative and nonpositive integers respectively by comparison of the general case to the special case in which a 1 and b c 0 and in which we can compute the limits exactly. We show that (T+ - ¿)-1 converges for ¿ +0 strongly to (T0+ - ¿)-1 if . In an analogous way, we define the operator T+, n (n in the Sobolev space H0-n(- 1, 1) as a restriction of t+ and prove strong convergence of (T+,n - ¿)-1 for ¿ +0 in this space of distributions if . With aid of the maximum principle we infer from this that, if h 1, the solution of t+u - ¿u = h, u(±1) = A ± B converges for ¿ +0 uniformly on [-1, - ] [, 1] to the solution of xu' - ¿u = h, u(±1) = A ± B for each p > 0 and for each ¿ if - .Finally we prove by duality that the solution of t-u - ¿u = h converges to a definite solution of the reduced equation uniformly on each compact subset of (-1, 0) (0, 1) if h is sufficiently smooth and if 1 - N.