Spectral properties of not necessarily self-adjoint linear differential operators

✍ Scribed by Bernd Schultze

• Publisher: Elsevier Science
• Year: 1990
• Tongue: English
• Weight: 828 KB
• Volume: 83
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1016/0001-8708(90)90069-y

✦ Synopsis

We consider not necessarily self-adjoint differential operators generated by ordinary differential expressions of the form My= f: pi(t)y"' on Z=[l, co) (*) i=O with n = ord(M) E N, pi E ci(Z, C). With A4 + we denote the adjoint expression b!f+y= i (-l)Qi(t)y)(i) i=O and with 7',(M) and T,(M) the minimal and maximal operator, respectively, generated by A4 in L2(Z). The basic spectral and extension theory (even in LP-spaces) was given by Rota [9]. In this theory the essential spectrum of M, o,(M) := (A E @ 1 range T,(M-A) is not closed}, plays a crucial role. If we assume that this set is not the entire plane, the following integers nul(M-A) := dim ker T,(M-A)

turn out to be constant (as functions of A) on each connected component of C\a,(M). These numbers are important because they indicate how many linearly independent boundary conditions one has to impose for a restriction of T,(M) (resp. an extension of T,(M)) in order to obtain a so-called maximal extension, i.e., an extension with minimal spectrum in this component. These maximal extensions correspond to the self-adjoint extensions in