What does $\textbf{H}(j\omega)$ mean physically?

This is the ``transfer function'' for a 4-terminal network. It's a complex function that describes how the input signal gets transformed (by whatever stuff is inside the box separating the two input terminals from the two output terminals) into an output signal, according to $\hat{V}_{ab} = \hat{H}(j \omega ) \hat{V}$, where $\hat{V}$ is the sinusoidal power supply voltage at the input and $\hat{V}_{ab}$ is what you see at the output. It's a function that describes the frequency response of the network: what you get out depends on the driving frequency $\omega$.