Unusual situations that arise with the Dirac delta function and its derivative

Abstract

There is a situation such that, when a function f(x) is combined with the Dirac delta function δ(x), the usual formula ∫∞-∞ f(x)δ(x)dx = f(0) does not hold. A similar situation may also be encountered with the derivative of the delta function δ′(x), regarding the validity of ∫∞-∞ f(x)δ′(x)dx = -f′(0). We present an overview of such unusual situations and elucidate their underlying mechanisms. We discuss implications of the situations regarding the transmission-reflection problem of one-dimensional quantum mechanics. © by the Sociedade Brasileira de Fsíca.