The function, which is NOT an eigenfunction of the indicated operator, is
Operator
Function
1. $\frac{d^{2}}{dx^{2}}-x^{2}$
$e^{\frac{-x^{2}}{2}}$
2. $\frac{d^{2}}{dx^{2}}+x^{2}$
$e^{\frac{-x^{2}}{2}}$
3. $\frac{d^{2}}{dx^{2}}$
$\cos\frac{\pi x}{4}$
4. $\frac{d^{2}}{dx^{2}}$
$e^{4ix}$
Answer
Answer: B ✓ checked by 4AB · confidence high
The source book printed no answer; this one was worked out and checked — see the explanation.
Explanation
For ψ = e^(−x²/2), ψ'' = (x² − 1)ψ. So (d²/dx² − x²)ψ = −ψ, an eigenfunction, but (d²/dx² + x²)ψ = (2x² − 1)ψ is not (option 2). cos(πx/4) and e^(4ix) are eigenfunctions of d²/dx².