Direct integral description of angulon

We propose a representation of angulon in which the angulon operator is decomposable relative to the field of Hilbert spaces over the probability measure space, and the probability measure corresponds to the total-number operator of phonons. In this representation we are able to find the system of $N+1$ equations whose solutions form the eigenspace of the angulon operator, where $1\leq N<\infty$ is the number of phonon excitations. Using this result we estimate the infimum of the spectrum. In the special case $N=1$, the lowest energy approximates to the value which is already known in the literature. Our findings indicate that two-phonon excitations ($N=2$) contribute notably to the energy of a molecule in superfluid $^4$He.