How to distinguish a mixture of two $d$-wave states from a pure $d$-wave state of high-temperature superconductors

By the study of the spectrum of collective modes in a pure d-wave state and in a mixed $d_{x^2-y^2}+id_{xy}$ state of high-temperature superconductors, we show that in spite of the fact that spectra in both pure states $d_{x^2-y^2}$ and $d_{xy}$ are identical, the spectrum in the mixture $d_{x^2-y^2} + id_{xy}$ state turns out to be quite different from them. Thus, the probe of the spectrum in ultrasound and/or microwave absorption experiments could be used to distinguish a mixture of two $d$-wave states from pure $d$-wave states.