TFPT scientific source excerpt
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000353 | \begin{theorem}[Primitive self-adjoint realization without extra analytic assumption]
000354 | \label{thm:well-posed-primitive-dynamics}
000355 | \label{thm:primitive-selfadjoint-realization}
000356 | Let $M_\Lambda$ be a finite cutoff of the primitive geometric anchor, so that $M_\Lambda$ is
000357 | compact with boundary. Let $D_{\mathrm{geo}}$ be the formally self-adjoint Dirac-type anchor and
000358 | let $A_\partial$ be an adapted boundary operator. Choose a self-adjoint $D_{\mathrm{geo}}$-elliptic
000359 | boundary condition $B_{\mathrm{sa}}$ of modified APS type:
000360 | \[
000361 | B_{\mathrm{sa}}=
000362 | \begin{cases}
000363 | H^{1/2}_{(-\infty,0)}(A_\partial), & \ker A_\partial=0,\\[2mm]
000364 | H^{1/2}_{(-\infty,0)}(A_\partial)\oplus L, & \ker A_\partial\neq0,
000365 | \end{cases}
000366 | \]
000367 | where $L\subset\ker A_\partial$ is chosen as in the normal form for self-adjoint $D$-elliptic
000368 | boundary conditions. Then the closed realization $D_{\mathrm{geo},B_{\mathrm{sa}}}$ is self-adjoint
000369 | Fredholm with discrete spectrum. If $D_{\mathrm{rel}}$ is a symmetric differential perturbation of
000370 | order at most one with bounded coefficients on $M_\Lambda$, then $D_{\mathrm{rel}}$ is
000371 | infinitesimally $D_{\mathrm{geo},B_{\mathrm{sa}}}$-bounded and therefore
000372 | \[
000373 | D_{\mathrm{geo},B_{\mathrm{sa}}}+D_{\mathrm{rel}}
000374 | \]
000375 | is self-adjoint on the same domain. In particular the finite-cutoff relative spectral seed is well
000376 | defined without an additional analytic existence hypothesis.
000377 | \end{theorem}