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?

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?