Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Table of Contents - 1999 - Volume 129, Issue 01  

Research Article

Examples of degenerate symmetric differential operators with infinite deficiency indices in L2(ℝm)

Yurii B. Orochkoa1

a1 Department of Applied Mathematics, Moscow State Institute of Electronics and Mathematics, Moscow 109028, Russia

For an unbounded self-adjoint operator A in a separable Hilbert space ℌ and scalar real-valued functions a(t), q(t), r(t), t ∊ ℝ, consider the differential expression

S0308210500027517_eqnU1

acting on ℌ-valued functions f(t), t ∊ ℝ, and degenerating at t = 0. Let Sp denotethe corresponding minimal symmetric operator in the Hilbert space S0308210500027517_inline1(ℝ) of ℌ-valued functions f(t) with ℌ-norm ∥f(t)∥ square integrable on the line. The infiniteness of the deficiency indices of Sp, 1/2 < p < 3/2, is proved under natural restrictions on a(t), r(t), q(t). The conditions implying their equality to 0 for p ≥ 3/2 are given. In the case of a self-adjoint differential operator A acting in ℌ = L2(ℝn), the first of these results implies examples of symmetric degenerate differential operators with infinite deficiency indices in L2(ℝm), m = n + 1.

(Received November 27 1996)