I have a basic question about quantum mechanics, maybe it has a basic answer.

Take a free particle in a quartic potential,

$L=\frac{1}{2}\dot{x}^2-\lambda x^4$

This is massless $\phi^4$ theory in 0+1 dimensions. Suppose that $\lambda$ is small. Then I want to know the spectrum of the theory. We expect a discrete spectrum with a small ground state energy. Is there a simple way to do perturbation theory here?

The reason I'm confused is because the free particle does not have a normalizable ground state. We could try to use ordinary first order perturbation theory, and the ground state energy is $\langle0|\lambda x^4|0\rangle$, but this is infinite. One other thing we could try is to regulate the free particle by adding a small mass $m$ or putting the theory in finite volume $L$. But then the answers only hold for $\lambda L^3\ll 1$ or $\lambda/m^3\ll 1$, so we can't remove the regulator completely.

For high energies we can use the WKB approximation, giving $E_n\sim \lambda^{1/3}n^{4/3}$. This doesn't work for the ground state though. In fact we could have guessed $E_n\sim \lambda^{1/3}$ by dimensional analysis. This alone suggests that perturbation theory is bad because the energies are nonanalytic in $\lambda$. (One can make the usual argument that the energies have to be nonanalytic in $\lambda$ by analytically continuing to negative $\lambda$ while analytically continuing the contour simultaneously.)

Are there any known approximation methods for this situation?

I have a basic question about quantum mechanics, maybe it has a basic answer.

Take a free particle in a quartic potential,

$L=\frac{1}{2}\dot{x}^2-\lambda x^4$

This is massless $\phi^4$ theory in 0+1 dimensions. Suppose that $\lambda$ is small. Then I want to know the spectrum of the theory. We expect a discrete spectrum with a small ground state energy. Is there a simple way to do perturbation theory here?

The reason I'm confused is because the free particle does not have a normalizable ground state. We could try to use ordinary first order perturbation theory, and the ground state energy is $\langle0|\lambda x^4|0\rangle$, but this is infinite. One other thing we could try is to regulate the free particle by adding a small mass $m$ or putting the theory in finite volume $L$. But then the answers only hold for $\lambda L^3\ll 1$ or $\lambda/m^3\ll 1$, so we can't remove the regulator completely.

For high energies we can use the WKB approximation, giving $E_n\sim \lambda^{1/3}n^{4/3}$. This doesn't work for the ground state though. In fact we could have guessed $E_n\sim \lambda^{1/3}$ by dimensional analysis. This alone suggests that perturbation theory is bad because the energies are nonanalytic in $\lambda$. (One can make the usual argument that the energies have to be nonanalytic in $\lambda$ by analytically continuing to negative $\lambda$ while analytically continuing the contour simultaneously.)

Are there any known approximation methods for this situation?

Edit: I figured out why there are no approximation methods for this theory. There is no small parameter! All the energies are proportional to $\lambda^{1/3}$, and the coefficients are pure numbers with no perturbation series. Maybe large N could help here.