What is an anisotropic harmonic oscillator? I can't find any explanation of it anywhere in the internet. How is it different from an isotropic harmonic oscillator? 
 A: In 3D, the harmonic oscillator potential can be characterized by
$$V(x) = \frac 1 2 m(\omega_x^2x^2 + \omega_y^2y^2 + \omega_z^2z^2).$$
If $\omega_x = \omega_y = \omega_z$, then the potential is spherically symmetric and we say that it is isotropic. Anisotropic simply means the opposite.
A: In two or three dimensions the possibilities for oscillation are considerable richer than in one dimension. In case isotropic harmonic oscillator for which the restoring force is proportional to the displacement from the equilibrium with the same constant proportionality factor in all directions F=Kx. F is the central force directed toward the equilibrium position.Where F(x)=kX, F(y)=kY, F(z)=kZ are the components of the force in all direction with same proportionality factor. 
but in the case of anisotropic oscillator The components are F(x)=k(x)X , F(y)=k(y)Y, F(z)=k(z)Z ; where the individual proportionality factor are all the different for all the individual components. Thats the actual basic difference between them.
An example of such force is the force felt by an atom displaced from its equilibrium position in a crystal of low symmetry where it experiences different force constants along the different axes
