  In this paper, we provide all possible characterisations of spectral measure
on the real line in terms of its two moments. To be more precise, we prove that
semispectral measure $F$ on the real line with compact support such that $T^p
=\int_{\mathbb{R}} x^p F(d x)$ and $T^q =\int_{\mathbb{R}}x^q F(dx)$, $p<q$,
$p$ odd and $q$ even, is spectral. Moreover we show that the set
$\Omega:=\{(p,q)\in \mathbb{N}^2\colon p<q, p \text{ odd and } q \text{
even}\}$ is best possible with respect to the above result, that is, if
$(p,q)\in \mathbb{N}^2\setminus\Omega$ there exist a semispectral measure $F$
on the real line which is not spectral and $T^p =\int_{\mathbb{R}}x^p F(dx)$
and $T^q =\int_{\mathbb{R}}x^q F(dx)$.
