For a potential of the form $V(x)=ax^4+bx^2$, where $a,b>0$, let us consider the one dimensional Schrodinger operator $D=-\frac{d^2}{dx^2}+V$ with Dirichlet B.C on $[-L,L]$ and denote its first eigenvalue by $\lambda(a,b)$.

**Q1.** Is $\lambda(a,b)$ a differentiable function in $a$ and $b$?

**Q2.** For which $(a,b)$'s subject to $a+b=1$, the function $\lambda$ is maximized/minimized?

yes, I believe, and a standard source for these things would be Kato's book. Also, you can make $V$ arbitrarily negative on $[-1/2,1/2]$, say, by taking $a\to\infty$, so $\lambda(a)$ will be unbounded below. $\endgroup$ – Christian Remling Nov 19 '19 at 17:59