I would like to find a function which graphically resembles $xe^{-x^2}$, or similarly the $\psi_{1}$ eigenfunction of the Schrodinger equation, and has the following properties:
It is continuous and differentiable. Also, the value at which the function converges to near-$0$ with a near-$0$ slope on the left and the right are fixed (say, to $x = 0$ and $x = 1$), and within the domain $[0,1]$, the area enclosed by each hump separated by the point m at which the function crosses the x-axis is equal. Since the area outside of the bounds is negligible, the domain restriction for area is not important. I am hoping to easily toggle where the point m lies, and even when it is closer to the left or right bound, still maintain the property that their areas are equal. Attached is a demos sketch that has disproportionate humps (move s to change the crossing point): https://www.desmos.com/calculator/0fnj3yxeru
and here is a partial solution, which is not differentiable at m: https://www.desmos.com/calculator/ywxezwhpk6?fbclid=IwAR0iWXLgRXcbz3swvUGjR8jUk847o0IlVPFb1v_07ENgXTW7hfB8Hf_Zl0E