We study the index of the APS boundary value problem for a strongly Callias-type operator $\mathcal{D}$ on a complete Riemannian manifold $M$. We use this index to define the relative $\eta$-invariant $\eta (\mathcal{A}_1 , \mathcal{A}_0)$ of two strongly Callias-type operators, which are equal outside of a compact set. Even though in our situation the $\eta$-invariants of $\mathcal{A}_1$ and $\mathcal{A}_0$ are not defined, the relative $\eta$-invariant behaves as if it were the difference $\eta (\mathcal{A}_1) - \eta (\mathcal{A}_0)$. We also define the spectral flow of a family of such operators and use it to compute the variation of the relative $\eta$-invariant.

中文翻译：

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具有非紧边界的偶数维流形的APS指数定理

我们研究了完全黎曼流形$ M $上强Callias型算子$ \ mathcal {D} $的APS边值问题的指数。我们使用该索引来定义两个强Callias型算子的相对$ \ eta $-不变$ \ eta（\ mathcal {A} _1，\ mathcal {A} _0）$，它们在紧凑集合外相等。即使在我们的情况下，未定义$ \ mathcal {A} _1 $和$ \ mathcal {A} _0 $的$ \ eta $不变式，相对的$ \ eta $不变式仍表现为差值$ \ eta（\ mathcal {A} _1）-\ eta（\ mathcal {A} _0）$。我们还定义了此类算子族的频谱流，并使用它来计算相对$ \ eta $-不变性的变化。