TFPT scientific source excerpt
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000001 | 
000002 | \begin{proof}
000003 | The existence of self-adjoint $D$-elliptic boundary conditions for formally self-adjoint Dirac-type
000004 | operators is standard. In the zero-kernel case APS is already self-adjoint. If the adapted
000005 | boundary operator has nontrivial kernel, one passes to a self-adjoint modified APS realization or,
000006 | equivalently, to
000007 | \[
000008 | H^{1/2}_{(-\infty,0)}(A_\partial)\oplus L,
000009 | \]
000010 | with $L\subset\ker A_\partial$ chosen as in the normal form theorem for self-adjoint $D$-elliptic
000011 | boundary conditions. The resulting realization is essentially self-adjoint on smooth sections
000012 | satisfying the boundary condition. Since $M_\Lambda$ is compact, coercivity at infinity is
000013 | automatic and the realization is Fredholm with discrete spectrum.
000014 | 
000015 | For the perturbation, standard first-order elliptic regularity on a compact manifold with
000016 | self-adjoint $D$-elliptic boundary condition gives a graph norm estimate
000017 | \[
000018 | \|\psi\|_{H^1}\le C\bigl(\|D_{\mathrm{geo},B_{\mathrm{sa}}}\psi\|+\|\psi\|\bigr).
000019 | \]
000020 | Because $D_{\mathrm{rel}}$ has order at most one and bounded coefficients,
000021 | \[
000022 | \|D_{\mathrm{rel}}\psi\|
000023 | \le
000024 | C_1\|\psi\|_{H^1}+C_0\|\psi\|
000025 | \le
000026 | \varepsilon\|D_{\mathrm{geo},B_{\mathrm{sa}}}\psi\|+C_\varepsilon\|\psi\|
000027 | \]
000028 | for every $\varepsilon>0$. Hence $D_{\mathrm{rel}}$ is $D_{\mathrm{geo},B_{\mathrm{sa}}}$-bounded
000029 | with relative bound zero. The Kato--Rellich theorem therefore yields self-adjointness of
000030 | $D_{\mathrm{geo},B_{\mathrm{sa}}}+D_{\mathrm{rel}}$ on the same domain.