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\\varepsilon_{i_{1}\\dots i_{n}}\n  \\int_{\\Lambda S_{n-1}}\\text{tr}_{\\mathbb{C}^d}\\, (U(x)(\\partial_{i_{1}}U)(x)\\dots\n  (\\partial_{i_{n-1}}U)(x)) x_{i_{n}}\\, d^{n-1} \\sigma(x), \\, (*) \\end{align} where $U(x) = \\text{sgn} \\,(\\Phi(x))$ and $L$ in $L^{2}(\\mathbb{R}^{n})^{2^{\\widehat n}d}$ is of the form \\[ L= 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Callias Index Formula Revisited","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Fritz Gesztesy, Marcus Waurick","submitted_at":"2015-06-16T20:37:11Z","abstract_excerpt":"We revisit the Callias index formula for Dirac-type operators $L$ in odd space dimension $n$, and prove that \\begin{align}\n  \\text{ind} \\, (L)\n  =\\bigg(\\frac{i}{8\\pi}\\bigg)^{\\frac{n-1}{2}}\\frac{1}{2(\\frac{n-1}{2})!}\n  \\lim_{\\Lambda \\to\\infty}\\frac{1}{\\Lambda }\\sum_{i_{1},\\dots,i_{n} = 1}^n \\varepsilon_{i_{1}\\dots i_{n}}\n  \\int_{\\Lambda S_{n-1}}\\text{tr}_{\\mathbb{C}^d}\\, (U(x)(\\partial_{i_{1}}U)(x)\\dots\n  (\\partial_{i_{n-1}}U)(x)) x_{i_{n}}\\, d^{n-1} \\sigma(x), \\, (*) \\end{align} where $U(x) = \\text{sgn} \\,(\\Phi(x))$ and $L$ in $L^{2}(\\mathbb{R}^{n})^{2^{\\widehat n}d}$ is of the form \\[ L= 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