Density of states in irregularly-shaped volumes

A very common result is that the density of momentum states in a cubic volume is $$\displaystyle\frac{V}{(2\pi\hbar)^3}$$ in momentum space. How does this result extend to arbitrary volumes? Are there any nice examples of volumes that are endowed with an interesting set of a momentum eigenstates?

• Semiclassically, it works for an arbitrary finite volume $V$ with a sufficiently nice boundary. Are you asking about exact quantum mechanical results? Commented Jan 14, 2020 at 13:32
• You might be interested to read about Weyl's formula: en.wikipedia.org/wiki/Weyl_law . This is basically the name by which this result for density of states is known for mathematicians. Commented Jan 14, 2020 at 14:37
• An interesting result from 1982 Density of states on fractals : “fractons” by S. Alexander & R. Orbach. Commented Jan 15, 2020 at 16:17
• @AleksandrArtemev that’s perfect, thanks! Commented Jan 16, 2020 at 22:06

Okay, so, for a free particle, with the hamiltonian $$H = - \frac{\hbar^2}{2m} \Delta$$ living in a cavity of volume $$V$$ (for any shape of the cavity) number of energy eigenstates with energy less than $$E$$ has the following asymptotic $$N(E) \sim \frac{V}{(2\pi \hbar)^d} \omega(\sqrt{2mE}),\, E \gg \frac{\hbar^2}{2ma^2}$$ Here $$\omega(R)$$ is a volume of $$d$$-dimensional ball of radius $$R$$, $$a$$ is a characteristic size of a cavity. This is a property of Laplacian eigenvalues called asymptotic Weyl formula, or "Weyl law": https://en.wikipedia.org/wiki/Weyl_law . By assuming $$E=p^2/2m$$, one gets the usually used density of states in momentum space $$\frac{V}{(2\pi \hbar)^d}$$. The formula is valid in any dimension.
The fact that this result is an asymptotic for large $$E$$ motivates this result being called "semiclassical". Unfortunately, I wasn't able to find any kind of "quantum-mechanical proof" to this formula that would derive it using semi-classical approximation; in physical literature it sometimes is loosely referred to as multidimensional analogue of Bohr-Sommerfeld rule, although I don't really feel this is the case, as Bohr-Sommerfeld rule speaks of a contour integral, rather than an integral over volume in momentum space.
The specifics of a cavity shape and boundary conditions become important for energies $$E \sim \frac{\hbar^2}{2ma^2}$$; the first correction term is proportional to the area of boundary of a cavity and has different signs for Dirichlet or Neumann b.c. . Maybe this correction term could be interpreted as some kind of "boundary states", though I am not sure about that.