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

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 $\left\{ {(\lambda ,\mu )|\omega _1 = \omega _2 } \right\} = \{ (\lambda ,\mu )|\mu = \frac{{\lambda ^2 }} {4}\} \subset C^2 .$