  In this work we study a class of anharmonic oscillators on $\mathbb{R}^n$
corresponding to Hamiltonians of the form $A(D)+V(x)$, where $A(\xi)$ and
$V(x)$ are
  $C^{\infty}$ functions enjoying some regularity conditions. Our class
includes fractional relativistic Schr\"odinger operators and anharmonic
oscillators with fractional potentials. By associating a H\"ormander metric we
obtain spectral properties in terms of Schatten-von Neumann classes for their
negative powers and derive from them estimates on the rate of growth for the
eigenvalues of the operators $A(D)+V(x)$. This extends the analysis in the
first part of our work, where the case of polynomial $A$ and $V$ has been
analysed.
