Operator Equations of Higher Order

Abstract

With the hypotheses and notation of the introductory section of Chapter V, we can write the general differential-operator equation ofmth order in the form

Lu ≡ (A0D^m _i+A₁D^m _i-1 +…+A_m) u=f.

Although, in principle, the method of Chapter V (under the hypothesis that A _k is a II-operator or an M-operator) is also directly applicable to (1), a number of technical points make it necessary to change the plan of the investigation in an essential way. It is enough to remark that already form> 2 there is no satisfactory (and form> 4, no) explicit representation for the roots of the characteristic equation

$\mathop \Sigma \limits_{j = {\text{0}}}^m {\text{ }}{{\text{A}}_{m - j}}{z^j} = {\text{0, }}{{\text{A}}_k} = {\text{const,}}$

in terms of the coefficients. This affects the method of using the solutions of the auxiliary ordinary equation of (1) (in which Afc are numbers) which arises in our discussion. Moreover, the boundary conditions of general form, defined by proper operators generated by the ordinary differential Operations (1) contain at least m2 essential parameters (§§2,3, Chapter III), and it has not been possible to obtain a sufficiently transparent characterization of the properties of the spectra of the corresponding operators for all possible choices of the boundary conditions.