An estimate for the resolvent of a non-self-adjoint differential operator on the half-line

作者：Gil M I^*

来源：Journal of Mathematical Physics, 2011, 52(4): 043515.

DOI：10.1063/1.3578931

We consider the operator defined by (T(0)y)(x) = -y ''+ q(x) y (x > 0) on the domain Dom (T-0) = {f is an element of L-2(0, infinity) : f '' is an element of L-2(0, infinity), f (0) = 0}. Here q(x) = p(x) + b(x), where p(x) and b(x) are real functions satisfying the following conditions: b(x) is bounded on [0,8), there exists the limit b(0) := lim(x ->infinity) b(x) and b(x) - b(0) is an element of L-2 (0, infinity). In addition, inf(x) p(x) > sup(x) vertical bar b(1)(x)vertical bar. We derive an estimate for the norm of the resolvent of T-0, as well as prove that (T-0 - ib(0)I)(-1) is a sum of a normal operator and a quasinilpotent one, and these operators have the same invariant subspaces.

• 出版日期2011-4