= Solution
Perturbation theory expands known <eigenvalues> and <eigenfunctions> about a solvable Hamiltonian. Small corrections require appropriate energy-gap and matrix-element scales; degeneracies must first be diagonalized within their subspace. The <variational method> instead minimizes a normalized trial state's <energy> expectation. It gives an upper bound on the ground <energy> without requiring a small perturbation; excited-state bounds need orthogonality constraints or an appropriate symmetry subspace.
Normalize the unperturbed oscillator states:
$$
|\psi_0|^2=\frac\alpha{\sqrt\pi}e^{-\alpha^2x^2},\qquad
|\psi_1|^2=\frac{2\alpha^3}{\sqrt\pi}x^2e^{-\alpha^2x^2}.
$$
For the <Gaussian perturbation of a quantum harmonic oscillator>, the first-order shifts are its expectation values:
$$
\boxed{\Delta E_0=\frac{V_0\alpha}{\sqrt{\alpha^2+\beta^2}},\qquad
\Delta E_1=\frac{V_0\alpha^3}{(\alpha^2+\beta^2)^{3/2}}.}
$$
For $\beta\gg\alpha$, they scale as $V_0\alpha/\beta$ and $V_0(\alpha/\beta)^3$ respectively: a narrow central perturbation couples weakly to the odd state's central <node>. For $\beta\ll\alpha$, both approach $V_0$ because the perturbation is nearly constant over the wavefunctions. The leading corrections are $V_0[1-\beta^2/(2\alpha^2)]$ and $V_0[1-3\beta^2/(2\alpha^2)]$.
For an improved variational estimate, replace the fixed width $\alpha$ by an adjustable positive width $s$ in each even/odd Gaussian. Their <energy> functionals are
$$
E_{\rm even}(s)=\frac{\hbar^2s^2}{4m}+\frac{m\omega^2}{4s^2}
+\frac{V_0s}{\sqrt{s^2+\beta^2}},
$$
$$
E_{\rm odd}(s)=\frac{3\hbar^2s^2}{4m}+\frac{3m\omega^2}{4s^2}
+\frac{V_0s^3}{(s^2+\beta^2)^{3/2}}.
$$
Minimize each separately. At $s=\alpha$ they recover the unperturbed <energy> plus the first-order shift, so optimization cannot raise those trial estimates. For this even one-dimensional potential the lowest odd state is the first <excited state>, making its odd-sector minimum an upper bound on $E_1$. Enlarging each parity-preserving trial basis can improve the estimates further.
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