I was wandering about an operator of the form $Lf=\sqrt{A^2+B^2\frac{d^2}{dx^2}}f$.
This would become $Lf\approx B\frac{df}{dx}$ if $B\gg A$ and $Lf\approx Af+\frac{B^2}{2A} \frac{d^2f}{dx^2}$ if $B\ll A$.
The idea would be to interpolate between a first order operator and a second order operator, somehow. However, I don't even know if this makes actual sense.
Have operators of this form been studied? Can they make sense?
EDIT--------------------------
Ispired by the comment about the Dirac equation.
Maybe I could write $L=\sum_{i\ge 0}a_i\frac{d^i}{dx^i}$, with constant coefficients $a_i$, and impose that $L^2=A^2+B^2\frac{d^2}{dx^2}$. This would give $a_0=A$, the condition $a_0a_1=a_1a_0$ would determine $a_1$, the condition $a_1^2+a_0a_2+a_2a_0=B^2$ would determine $a_2$ and so on.
Such relations could only be satisfied by matrix-valued coefficients, but be that as it may. I would then have an infinite series for the operator $L$, which could in principle be truncated to a desired approximation.
Does this make sense?
Since $A$ and $B$ might be negative we need to compare the absolute values
$$Lf=\sqrt{A^2+B^2\frac{d^2}{dx^2}}f\approx |B|\sqrt{\frac{d^2f}{dx^2}}$$
$$Lf=\sqrt{A^2+B^2\frac{d^2}{dx^2}}f=A\left( 1+\frac{B^2}{A^2}\frac{d^2}{dx^2}\right)^\frac12f \approx Af+\frac{B^2}{2A}\frac{d^2f}{dx^2}$$