Exterior Dirichlet parametrix and compact decaying potentials

Write ${\mathrm{\Sigma }}_0$ for the end component and $M_1,\dots ,M_m$ for the finitely many compact components supplied by [F4]. The spaces $X_0,Y_0$ on ${\mathrm{\Sigma }}_0$ are Banach by [F3], Proof 4.1, and the compact-component spaces $X_j,Y_j$ are Banach by [F4], Proof 2.1, at orders four and two respectively. Here $X_j$ has zero trace when $\mathrm{\partial }{M}_j\mathrm{\ne }\mathrm{\varnothing }$. Restriction identifies $X_{\beta }$ and $Y_{\beta }$ with these finite direct sums, with equivalent norms: the annular term involves only ${\mathrm{\Sigma }}_0$, and the compact terms are finite chart norms. Smooth coordinate transitions compare these compact norms by the chain and product rules. Restriction decreases each chart supremum and seminorm; conversely, pairs lying in different components of a fixed compact chart subset have positive Euclidean separation, so their Hölder quotients are bounded by derivative suprema. Thus the full norm and the sum of the component norms bound each other by fixed constants. A Cauchy sequence in either full space is therefore Cauchy in each component space, has component limits there, and converges in the finite sum norm and hence in the original norm. The component limits glue across the disjoint open-and-closed components and preserve the zero boundary values. This proves completeness of both full-exterior spaces.

Let $H_j$ denote the mean-zero inverse of ${\mathrm{\Delta }}_h$ on a closed component $M_j$. The estimate $\mathrm{\parallel }{H}_{j}f{\mathrm{\parallel }}_{C^{4,\alpha }(M_j)}\le C_j\mathrm{\parallel }f{\mathrm{\parallel }}_{C^{2,\alpha }(M_j)}$ for mean-zero $f$ is proved in [F4], Proof 4.1; it is not inferred merely from a kernel-and-range assertion. Put $P_{j}f={\mu }_h(M_j{)}^{-1}{\int }_{M_j}fd{\mu }_h$, viewed as a constant function. Since $\mathrm{\mid }{P}_{j}f\mathrm{\mid }\le \mathrm{\parallel }f{\mathrm{\parallel }}_{C^0(M_j)}$, this is a bounded projection at both relevant Hölder orders. Use $H_j(I-P_j)$ as the closed-component block of $G$, the bounded compact Dirichlet inverse on each nonempty-boundary compact component, and the bounded end inverse on ${\mathrm{\Sigma }}_0$. The finite sum and step 1.1 make $G:Y_{\beta }\to X_{\beta }$ bounded. The same componentwise reasoning makes ${\mathrm{\Delta }}_h:X_{\beta }\to Y_{\beta }$ bounded. The inverse identities on each component give ${\mathrm{\Delta }}_{h}G=I_{Y_{\beta }}-{\mathrm{\Pi }}_Y$ and $G{\mathrm{\Delta }}_h=I_{X_{\beta }}-{\mathrm{\Pi }}_X$, where ${\mathrm{\Pi }}_X,{\mathrm{\Pi }}_Y$ apply $P_j$ on closed components and vanish elsewhere. They are bounded finite-rank projections.

Let $q$ be the number of closed compact components. Their constants are in the Laplacian kernel, and $G{\mathrm{\Delta }}_h=I-{\mathrm{\Pi }}_X$ implies that these are the entire kernel. The closed-component range assertion in [F4] gives ${\mathrm{\Pi }}_Y{\mathrm{\Delta }}_h=0$, while ${\mathrm{\Delta }}_{h}G=I-{\mathrm{\Pi }}_Y$ shows every element of $\ker {\mathrm{\Pi }}_Y$ lies in the range. Hence $\mathrm{ran}{\mathrm{\Delta }}_h=\ker {\mathrm{\Pi }}_Y$, a closed subspace. The kernel and cokernel both have dimension $q$, since ${\mathrm{\Pi }}_X$ and ${\mathrm{\Pi }}_Y$ project onto the $q$ independent closed-component constants. Together with boundedness and the Banach property this proves Fredholmness and index zero, with all inner-boundary conditions retained.

Multiplication by $V$ is a compact bounded linear map by [F5]. Step 1.1 establishes the Banach hypotheses on the full exterior, including its compact components, and step 3.1 establishes Fredholmness of the bounded Dirichlet Laplacian. Apply [F6] to this operator and the compact map $-M_V$, where $M_{V}u=Vu$. The operator ${\mathrm{\Delta }}_h-V$ is Fredholm with the same index zero. No sign assumption on $V$ was used. This proves all assertions.