Static and dynamic properties of Josephson weak links with singlet and triplet coupling

We theoretically study static and dynamic properties of short Josephson junctions (JJ) with singlet and triplet Josephson coupling. In singlet Josephson weak links, two singlet superconductors S are connected with each other by a normal film (N) or wire. Triplet JJs, which we denote S$_{\text{m}}$/N(F)/S$_{\text{m}}$, are formed by two singlet BCS superconductors covered by a thin layer of a weak ferromagnet F$_{\text{w}}$. These superconductors S$_{\text{m}}$ are separated from the N (or F) layer by spin filters, which pass electrons with only one spin orientation. The triplet Cooper pairs propagating from the left (right) superconductors S$_{\text{m}}$ differ from each other not only by polarizations, but also by chiralities. The latter is determined by the magnetization orientation in weak ferromagnets F$_{\text{w}}$. We obtain analytical formulas for the critical Josephson current in both types of the JJs. If chiralities of the triplet Cooper pairs penetrating into the N film in S$_{\text{m}}$/N(F)/S$_{\text{m}}$ JJs from the left and right S$_{\text{m}}$ are different, the Josephson current is not zero in the absence of the phase difference (spontaneous Josephson current). We also calculate the admittance $Y(\Omega)$ for arbitrary frequencies $\Omega$ in the case of singlet JJs and for low frequencies in the case of triplet JJs. At low temperatures $T$, the real part of the admittance $Y^{\prime}(\Omega)$ in singlet JJs starts to increase from zero at ${\hbar \Omega \geq \Delta_{\text{sg}}}$, but at ${T \geq \Delta_{\text{sg}}}$, it has a peak at low frequencies the magnitude of which is determined by inelastic processes. The subgap $\Delta_{\text{sg}}$ depends on transparencies of the S/N interfaces and on the phase difference $2 \chi_{0}$. The low-frequency peak in $Y^{\prime}(\Omega)$ in triplet JJs disappears.