= Gaussian variational bound for a radial harmonic oscillator
{c}
{title2=$\lambda_0=n,\quad y_0=e^{-x^2/2}$}
Consider $-y''-(n-1)y'/x+x^2y=\lambda y$ for a positive <integer> $n$, with regular radial data at zero and square-integrable decay at infinity in the measure $x^{n-1}dx$. Its <Sturm-Liouville form> has $p=w=x^{n-1}$ and $q=x^{n+1}$. For the Gaussian trial $y=e^{ax^2}$, $a<0$, the <Rayleigh quotient> is
$$
\Lambda(a)=(4a^2+1)\frac{\int_0^\infty x^{n+1}e^{2ax^2}dx}{\int_0^\infty x^{n-1}e^{2ax^2}dx}=-n\left(a+\frac1{4a}\right).
$$
The <Gamma function recurrence> evaluates the ratio, and the minimum is $n$ at $a=-1/2$. It is the exact lowest <eigenvalue>: after the endpoint term $[x^ny^2]_0^\infty$ vanishes, <integration by parts> gives
$$
\int_0^\infty\left[x^{n-1}(y')^2+x^{n+1}y^2\right]dx-n\int_0^\infty x^{n-1}y^2dx=\int_0^\infty x^{n-1}(y'+xy)^2dx\geq0.
$$
Equality gives the displayed positive Gaussian <eigenfunction>. For $n=1$, the radial regularity condition is the <Neumann boundary condition> at zero, not a vanishing value there. Changing the boundary condition changes the lowest <eigenvalue> and whether this trial is admissible.
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