**I.** Classical case. Let us have a particle with the initial position $x(0)$ and the initial velocity $v(0)$ with the corresponding uncertainties denoted with corresponding $\delta_x$ and $\delta_v$. For simplicity, let us consider a free motion. Then the solution is known: $$x(t) = x(0)\pm \delta_x +v(0)t\pm \delta_v t\qquad(1).$$ Thus, the position **absolute **uncertainty $\delta x(t)$ grows with time as $\pm\delta_v t$. Now, if we put this particle in a box $x\in [0,L]$, then starting from a certain time the particle position uncertainty will become larger than the box size and the particle position will become completely uncertain within the box. All we may say is that $x\in[0,L]$ with a constant probability distribution $f(X)=1/L$.

Something similar I expect from a classical linear oscillator "dynamics": its position uncertainty becomes limited with the turning points with some oscillator-like probability density between them.

**II. QM. **So I jump to the quantum mechanical case where you propose to play with the linear oscillator frequency (energy) via "better" choice of $k(g)$. Let us admit that we managed to guess the function $k(g)$ giving the exact eigenvalue $E_0(g)$ with help of a “free oscillator”. Using $\tilde{E}^{(0)}_0(g) = E_0(g)$ does not prevent us from unnecessary perturbative eigenvalue corrections (I called such series “blank”): $W_{00}= \langle [k(g)-k]x^2/2+g x^4/4\rangle_{00}\ne 0$, $W_{nm}\ne 0$. The whole series starting from the “first order” $W_{00}$ results in principle (i.e., when correctly summed up) into zero since no corrections to $\tilde{E}^{(0)}_{00}(g)= E_0(g)$) should exist, but each term in each order is not zero since $k(g)$ is a non trivial function of $g$ and one does not expands it in powers of $g$. In other words, a truncated perturbative series is a non-zero-valued polynomial. I encountered such a case too, many years ago. Despite such a series is unnecessary for the eigenvalue (it spoils the good initial approximation), the corresponding series for the eigenfunction is useful since it modifies a wrong linear oscillator wave function into a anharmonic oscillator wave function. I cannot say what a long-time dynamics can be predicted for a superposition of such states.

By the way, in my practice I encountered only one one case when knowing the exact eigenvalue determined the exact eigenfunction: it was the case of piece-wise behaviour of the Sturm-Liouville problem coefficients (a multi-layer system). Then one can write exactly (analytically) the solution within each layer, something like this: $\psi_E(x)=A_i \sin[\sqrt{E} (x-x_i)]+B_i \cos[\sqrt{E}(x-x_i)], \; i=1,2,3,…$, where $i$ is the layer number. Injecting the exact eigenvalue in this analytical formula gives the exact eigenfunction, but for that to be possible, one has to be able to express the eigenfunction as an analytical function of $E$. In particular, I used it for describing a long-time dynamics of heat conduction of a multi-layer body.

If one uses the regular perturbation theory, one obtains here a series with the zero-radius of convergency. For summing (partially) its terms, many approaches have been proposed (including mine ;-), see also here (**Отступление.)**). I am not an expert in this field, but my practice shows that any available information about the exact solution may help construct reasonable approximations.