Factorization of degenerate Cauchy problems: The linear case. (English) Zbl 0881.47021

Thaller, B.; Thaller, S.

J. Oper. Theory 36, No. 1, 121-146 (1996).

Summary: We investigate an abstract degenerate Cauchy problem with a non-invertible operator \(M\) at the derivative. The problem is formulated in a Hilbert space \(\mathfrak H\) which can be written as an orthogonal direct sum of \(\text{Ker }M\) and \(\text{Ran }M^*\). Under certain conditions it is possible to reduce the problem to an equivalent non-degenerate Cauchy problem in the factor space \({\mathfrak H}/\text{Ker }M\). The explicit form of the generator for the restricted problem is investigated. As an example, we discuss the Dirac equation, where our theory leads to a new interpretation of the nonrelativistic limit. We show that this limit can be understood in terms of a degenerate Cauchy problem where the generator of the restricted problem is a Schrödinger operator. Finally, we describe some consequences for the treatment of degenerate control systems. In particular, we introduce dual methods of factorization in order to investigate the compatibility of factorization and transition to the dual system, which is essential for the definition of basic notions like observability and detectability.

MSC:

47D06	One-parameter semigroups and linear evolution equations
93B28	Operator-theoretic methods
35Q40	PDEs in connection with quantum mechanics

Keywords:

control theory; abstract degenerate Cauchy problem; non-invertible operator; orthogonal direct sum; Dirac equation; Schrödinger operator; methods of factorization; observability; detectability