Connection between spectral and oscillatory properties of differential- operator equations of an arbitrary order. II: Applications of factorization and squaring. (Russian) Zbl 0667.34032
This paper is the continuation of Teor. Funkts., Funkts. Anal. Prilozh. 48, 49-60 (1987; Zbl 0645.34024). The principal result, here, is the following:
Theorem 3. Let \(\omega (x)\in B(H^ n,H)\) be a solution of the equation \[ \ell [y]\equiv \sum^{n}_{j=0}(-1)^ jD^ jp_ j(x)\cdot D^ jy+\frac{i}{2}\sum^{n-n-s}_{j=0}(-1)^ jD^ j\{Dq_ j(x)+q^*_ j(x)D\}D^ jy=\lambda W(x)y, \] D\(=\frac{d}{dx}\), for \(\lambda\in 0\), \(r=2n\). Then in every point \(x\in (a,b)\) where \((\omega^{\lambda})^{- 1}\in B(H^ n)\) exists, the operation \(\ell\) can e represented in the multiplicative form \(\ell [y]=\mu^*p_ n(x)\mu [y]\), where \(\mu [y]=y^{(n)}-\omega^{(n)}(x)\). \((\omega^{\lambda})^{- 1}y^{\lambda}\), \(\mu^*\) adjoint operation of \(\mu\) in \(L_ 2(H_ j(a,b),dx)\) and also \(\mu (\omega)=0\), \(\mu^+[\{0_{H'};...;O_{H'};I_ H\}(\omega^{\lambda})^{*- 1}]=0.\) This theorem is a generalization of a Frobenius factorization theorem.
Theorem 3. Let \(\omega (x)\in B(H^ n,H)\) be a solution of the equation \[ \ell [y]\equiv \sum^{n}_{j=0}(-1)^ jD^ jp_ j(x)\cdot D^ jy+\frac{i}{2}\sum^{n-n-s}_{j=0}(-1)^ jD^ j\{Dq_ j(x)+q^*_ j(x)D\}D^ jy=\lambda W(x)y, \] D\(=\frac{d}{dx}\), for \(\lambda\in 0\), \(r=2n\). Then in every point \(x\in (a,b)\) where \((\omega^{\lambda})^{- 1}\in B(H^ n)\) exists, the operation \(\ell\) can e represented in the multiplicative form \(\ell [y]=\mu^*p_ n(x)\mu [y]\), where \(\mu [y]=y^{(n)}-\omega^{(n)}(x)\). \((\omega^{\lambda})^{- 1}y^{\lambda}\), \(\mu^*\) adjoint operation of \(\mu\) in \(L_ 2(H_ j(a,b),dx)\) and also \(\mu (\omega)=0\), \(\mu^+[\{0_{H'};...;O_{H'};I_ H\}(\omega^{\lambda})^{*- 1}]=0.\) This theorem is a generalization of a Frobenius factorization theorem.
Reviewer: I.Onciulescu
MSC:
| 34L99 | Ordinary differential operators |
| 34C10 | Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations |
| 47E05 | General theory of ordinary differential operators |
