Let $H$ be a Hilbert space with orthonormal basis $\mathscr{E}$ and $g:\mathscr{E}\rightarrow \mathbb{C}$ be bounded. Define the operator $M|x\rangle =\sum_\mathscr{E} g(e) |e\rangle \langle e|x \rangle$ where I'm using the braket notation for inner products and the summation is regarded as nets. Then it's clear that $\mathscr{E}$ are the eigenstates of operator $M$ with eigenvalue $g(e)$.
Now consider a similar setup. Let $g\in L^\infty$ and $M:L^2 \rightarrow L^2$ be the operator defined by $Mf=gf$. Then $M$ does not have eigenvectors in $L^2$. However, if $|x\rangle =\delta_x$ represents the Dirac distribution shifted at $x$, i.e., $\langle x|f \rangle =f(x)$, then it's clear that $|x \rangle$ are "eigenvectors" of $M$ with eigenvalues $g(x)$. Is there a formulation of such "generalized eigenvectors"?
There is Rigged Hilbert space but mathematical physicists don't use it much.