Where in experiment do you encounter Lorentizan wavefunction? Is there an experimental system, or such that can be observed in nature where a particle's wave function assumes a form - $\psi(x)\propto \frac{1}{\sqrt{x^2+1}}$ such that $|\psi(x)|^2$ is Lorentzian?
An answer to this question Is there a condition of quantum mechanics that forbids Lorentzian distributions? claims that such wavefunctions are responsible for charge delocalization in molecules. I could not find a reference that discusses this. 
My ultimate goal here is to understand whether quantities like $<\hat{X}>$ would assume finite values or indefinite values (since a Lorentzian  distributions has no moments).
 A: $\psi(x) = \frac{1}{\sqrt{1+x^2}}$ is a perfectly valid wavefunction.  However, one cannot find the expected value of $\hat X$ in this state, because $\psi$ is not in the domain of the $\hat X$ operator.
The appropriate domain on which $\hat X$ is self-adjoint is
$$D_X:= \left\{\psi \in L^2(\mathbb R) \ \left| \ \int_\mathbb R |x \psi(x)|^2 dx < \infty\right\}\right.$$
For all $\psi \in D_X$, we then have that $(\hat X \psi) (x) = x\psi(x)$.  However, since you can see pretty quickly that  $\int_\mathbb R |x\psi(x)|^2 dx \rightarrow \infty$ for $\psi(x) = 1/\sqrt{1+x^2}$, this wavefunction is not in the domain of the position operator, which makes 
$$\langle \hat X\rangle_\psi \equiv \frac{\langle \psi,\hat X \psi\rangle}{\langle\psi,\psi\rangle}$$
undefined.  There's nothing particularly weird about this.  What it means is as follows: if you measure the position of each of $N$ identically prepared systems and denote the average as $\overline X_N$, then as $N\rightarrow \infty$, $\overline X_N$ will not converge to a finite value.  If you wait long enough, you'll measure arbitrarily large positions, and this will occur sufficiently regularly that the overall average is not bounded as the number of trials gets larger.
Another way to think of it is that, given any $M>0$, you will measure your overall ensemble average position to be greater than $M$ (or less than $-M$) with probability 1 as the number of measurements tends to infinity.
A: One potential example of an experimental system that has this type of Lorentzian "wavefunction" is cavity-enhanced parametric down conversion (PDC) photon sources.
In this case it isn't the position wavefunction  but the "wavefunction" in frequency space which is Lorentzian (more correctly the joint spectral intensity (JSI) of two photons is an Airy distribution but individual cavity modes are Lorentzian). This JSI is like the probability density function (PDFs) for two photons to be at frequencies $\omega_1$ and $\omega_2$ but the marginals describe single particle PDFs.
If you want to read more, I think this is a nice theory paper. 
A: In the post you mention there are some references. Also, they tackle your question about the interpretation of expected values. Nevertheless I think you are interestred in wavefunctions $\psi$ such that $\vert\psi\vert^2$ is a Lorentzian, which, you know, is not eactly the same! 
Edit: Sorry to say that Loretzian wave-function is not square integrable. 
