二阶非自伴微分算子的特征展开

THE EIGENFUNGTION EXPANSION THEOREMFOR THE SECOND-ORDER NON-SELFADJOINTDIFFERENTIAL OPERATER

目的:对二阶非自伴微分算子l(y)=-y″+q(x)y,0≤x<∞,在假定integral from 0 to ∞(x^2|q(x)|dx)<∞的条件下证明了特征展开。本文对方程的解做了更精确估计,然后直接采用回路积分,这样不但简化展开式的证明,且对q(x)的要求可降低为integral from 0 to ∞(x|q(x)|dx)<∞。

M. A. Haap has proved the eigenfunction expansion theorem, for the second-order non-selfadjoint differential operator l(y)= -y" + q(x)y under theconditionx2| q(x) | dx<. Using contour integral, we have not only simplifiedthe proof, but also replaced his conditipn by the weaker one x|q(x)|dx<.