# Well-posedness of boundary-value problems for equation (1) in the case of commuting self-adjoint A and B

• Alexander Ya. Shklyar
Chapter
Part of the Operator Theory Advances and Applications book series (OT, volume 92)

## Abstract

Let −∞ < a < b ≤ +∞.

For each (λ,µ) ∈ C2 denote by ψ i (λ,µ,t) (t ∈ [a,b]), i = 0,1, solutions of the scalar ordinary differential equation (o.d.e.)
$$u''(t) + \lambda u'(t) + \mu u(t) = 0$$
(2.1)

On [a, b] such that $${\psi _0}(\lambda ,\mu ,a) = 1,{\psi '_0}(\lambda ,\mu ,a) = 0,{\psi _1}(\lambda ,\mu ,a) = 0,{\psi '_1}(\lambda ,\mu ,a) =$$

Here for any t ∈ [a, b]: $${\psi _0}(\lambda ,\mu ,t) = \frac{{{\omega _1}{e^{{\omega _2}(t - a)}} - {\omega _2}{e^{{\omega _1}(t - a)}}}}{{{\omega _1} - {\omega _2}}},$$$${\psi _1}(\lambda ,\mu ,t) = \frac{{{e^{{\omega _1}}}^{(t - a)} - {e^{{\omega _2}}}^{(t - a)}}}{{{\omega _1} - {\omega _2}}}$$

For (λ,µ) ∈ C2 such that ω1≠ω2, and ψ i (λ,µ,t), i = 0,1, can be extended by continuity to the set $$\left\{ {(\lambda ,\mu )|\omega _1 = \omega _2 } \right\} = \{ (\lambda ,\mu )|\mu = \frac{{\lambda ^2 }} {4}\} \subset C^2 .$$