A general comparison theorem

Using the Hellmann-Feynman theorem, a general comparison theorem is established for an eigenvalue equation of the form $(T+V)\mathrm{\mid }\psi >=E\mathrm{\mid }\psi >$, where $T$ is a kinetic part which depends only on momentums and $V$ is a potential which depends only on positions. We assume that $H^{(1)}=T+V^{(1)}$ and $H^{(2)}=T+V^{(2)}$ ($H^{(1)}=T^{(1)}+V$ and $H^{(2)}=T^{(2)}+V$) support both discrete eigenvalues $E_{\{\alpha \}}^{(1)}$ and $E_{\{\alpha \}}^{(2)}$, where $\{\alpha \}$ represents a set of quantum numbers. We prove that, if $V^{(1)}\le V^{(2)}$ ($T^{(1)}\le T^{(2)}$) for all position (momentum) variables, then the corresponding eigenvalues are ordered $E^{(1)}{{\alpha}} \le E^{(2)}{{\alpha}}$. Some analytical applications are given.