Consider the one dimensional Schrödinger hamiltonian $\mathcal{H}=-\frac{\hbar^2}{2} \frac{d^2}{dx^2} + V(x)$.

Suppose that $V:\mathbb{R} \rightarrow \mathbb{R}^+$ is a continuous and confining potential $\displaystyle \lim_{\lvert x\rvert \to +\infty} V(x)=+\infty$

It is well known that $\mathcal{H}$ has pure discrete spectrum $\lambda_1< \cdots \leq \lambda_n$ with $\lambda_n \to +\infty$ as $n\to + \infty$.

Furthermore, if $V$ is a polynomial of even degree $2s$, then the eigenvalues obey the asymptotic formula

$$\lambda _k \sim {C_{s,\hbar}\, k}^{\frac{2s}{s+1}}$$

**I want to know if the above asymptotic formula holds if**, instead of having a polynomial potential as above, I consider a "pseudo-polynomial" in the sense that $V$ is another function on a segment $[a,b]\subset \mathbb{R}$ (like $\sin(x)$) and outside $\mathbb{R}\setminus[a,b]$ its an even polynomial such that the result is a continuous piecewise function.