Regularity of spectral problems with supplementary conditions at interior points (Q1814562)

Consider the spectral problem \[ \ell(y,\lambda)=y^{(n)}+p_ 1(x,\lambda)y^{(n-1)}+\dots+p_ n(x,\lambda)y=0, \leqno(1) \] \[ U_ j(y,\lambda)=\sum^{s_ 1}_{s=0}\sum^{r_ 1}_{r=0}(a^{j0}_{rs}\lambda^ ry^{(s)}(0)+a^{j1}_{rs }\lambda^ ry^{(s)}(1))=0,\qquad 1\leq j\leq n, \leqno(2) \] \[ V_{\nu t}(y,\lambda)=\sum^{s_{\nu t}}_{s=0}\sum^{r_{\nu t}}_{r=0}(b^{\nu t}_{rs}\lambda^ ry^{(s)}(\alpha_ t- 0)+d^{\nu t}_{rs}\lambda^ ry^{(s)}(\alpha_ t+0))=0,\qquad 1\leq\nu\leq n, \leqno(3) \] \(1\leq t\leq m-1\), \(r_ j+s_ j\leq n-1\), \(0<\alpha_ 1<\dots<\alpha_{m-1}<1\), \(r_{\nu t}+s_{\nu t}\leq n-1\), \(p_ s(x,\lambda)=\sum^ s_{l=0}p_{ls}(x)\lambda^ l\), \(p_{ss}\)=const, \(p_{nn}\neq 0\), \(a^{j0}_{rs}\), \(a^{j1}_{rs}\), \(b^{\nu t}_{rs}\), \(d^{\nu t}_{rs}\) are complex constants. The spectral problem (1)--(3) is called regular if every coefficient \(p_{ls}(x)\) in (1) is Lebesgue integrable and \(F^ s\neq 0\) in the characteristic determinant \[ \Delta(\lambda)=\lambda^ \kappa(\sum^ h_{s=1}[F^ s]\exp(\lambda_{\mu_ s})+\sum_ r[A^ r]\exp(\lambda\beta_ r)), \] where \([\eta]=\eta+O(\lambda^{-1})\), \(\eta\)=const. Then the Green function of the spectral regular problem (1)-(3) satisfies \(| G(x,\xi,\lambda)|<C|\lambda|^{1- n}\) in the whole complex plane except the disks of radius \(\varepsilon\) and centered in the eigenvalues of the problem.