Consider $V(x)$ a one dimensional polynomial, confining, symmetric double well potential i.e.

- $V(x)=V(-x)$ for all $x\in\mathbb{R}$
- $\displaystyle \lim_{x\to\pm\infty} V(x)=+\infty$
- $V(x)\in \mathbb{R}_{2n}[x]$
- $V$ has two local minima $x_1,x_2$ such that $x_1=-x_2$.

Then the operator $H=-\frac{d^2}{dx^2}+V(x)$ has pure discrete spectrum $(\lambda_i)_{i\geq0}$ and associated to each eigenvalue there is an eigenfunction $\phi_i$.

**My questions are:**

I read that there are 2 eigenfunctions $\psi_1,\psi_2$ , associated to "each well of the potential" (localisation in one side or the other). What is the relation between those "more elementary eigenfunctions" $\psi_1,\psi_2$ and the ground state $\phi_0$. I understant that is a sort of linear combination, but, why we cannot say that it's $\psi_1$ the ground state instead of $\phi_0$?

With the above assumptions on the potential, the question asked here has a positive answer? Or can we at least make explicit the growth of a bound $M_k$ depending on $k$? i.e. $\lvert\lvert \phi_k \rvert \rvert_\infty \leq M_k $ and $M_k\sim k^{\rm something}$

Thanks!