= Exponential-polynomial trial space for a mixed Sturm-Liouville problem
{title2=$y_n(x)=x^ne^{-nx},\quad y_n(0)=y_n'(1)=0$}
For $-(p y')'=\lambda w y$ on $[0,1]$ with $p=(1+x)e^{2x}$, $w=e^{2x}$, $y(0)=0$ and $y'(1)=0$, the <Rayleigh quotient> is $\int p y'^2/\int w y^2$. The trial $y_1=xe^{-x}$ gives numerator $5/12$ and denominator $1/3$, hence $0<\lambda_{\min}\leq5/4$. Every displayed $y_n$ obeys the same mixed <boundary conditions>; so does $\sin[(m+1/2)\pi x]$. Adding any of these to the trial space cannot worsen the optimized upper bound. Matrix elements $A_{ij}=\int p y_i'y_j'$ and $M_{ij}=\int w y_iy_j$ give a finite-dimensional <generalized eigenvalue problem> $Av=\lambda Mv$. The case $n=1$ repeats the original trial and adds no variational freedom.
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