= Solution
Factorizing the projector found in part <S18/2/b> gives the normalized <eigenfunctions>
$$
\boxed{\varphi_n(x)=\sqrt{\frac2a}\sin\frac{n\pi(x+a/2)}a,
\quad -a/2<x<a/2,\quad n\ge1,}
$$
with zero continuation outside. Their squared <integrals> are one and different positive integers give orthogonal functions. These are the familiar infinite-square-well states shifted so that the well centre is at zero. Each <eigenfunction> has an arbitrary overall constant <phase>, which cancels from the projector.
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