Finite gapped spectral projection regularity

Fix $s_0\in [0,1]$. By [F1], the selected finite set of eigenvalues at $s_0$ is separated from its complement by the positive gap. Choose finitely many positively oriented closed contours whose interiors contain exactly this cluster and whose boundaries avoid the spectrum; for a sufficiently small neighborhood of $s_0$, the resolvent on these contours remains defined. Indeed, if $R_0(z)=(H(s_0)-zI{)}^{-1}$ and $\mathrm{\parallel }H(s)-H(s_0)\mathrm{\parallel }\mathrm{\parallel }{R}_0(z)\mathrm{\parallel }<1$ uniformly on the compact contour, then submultiplicativity [F3] makes the Neumann series for $I+(H(s)-H(s_0))R_0(z)$ convergent and gives the inverse.

On this neighborhood, [F2] and the scalar residue formula give the same Riesz projection as the definition: ${\displaystyle P(s)=-\frac{1}{2\pi i}{\oint }_{\mathrm{\Gamma }}(H(s)-zI{)}^{-1}dz\mathrm{.}}$ To see this, diagonalize $H(s)$; the integral of $(\lambda -z{)}^{-1}$ is $-2\pi i$ when $\lambda$ is inside the contour and zero when it is outside, so the sum is exactly the projection onto the selected eigenspaces.

Put $R_z(s)=(H(s)-zI{)}^{-1}$. Differentiating $(H-zI)R_z=I$ gives ${\dot{R}}_z=-R_z{H}^{\mathrm{\prime }}{R}_z$ and differentiating again gives ${\ddot{R}}_z=2R_z{H}^{\mathrm{\prime }}{R}_z{H}^{\mathrm{\prime }}{R}_z-R_z{H}^{\mathrm{\prime }\mathrm{\prime }}{R}_z$. The contour is compact and stays uniformly in the resolvent set on this neighborhood, so these integrands and their first two derivatives are continuous uniformly in $z$. Differentiation under the finite contour integral is therefore valid, and step 2.1 makes $P$ twice continuously differentiable near $s_0$.

On the overlap of two such neighborhoods, each contour formula projects onto the same coherently followed eigenspaces by [F1]–[F2]; hence the local projections agree. Their first two derivatives consequently agree on overlaps, so the local $C^2$ representations patch to a global $C^2$ map on $[0,1]$, with one-sided derivatives at the endpoints.