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Sign of a perturbed massive Dirac operator and associated Fredholm module. (English) Zbl 1414.35191

Summary: We provide an algorithm to compute the sign of the massive Dirac operator \(D_m\), \(m > 0\), on \(\mathbb{R}^d\), \(d \geq 2\), perturbed by an electromagnetic potential \(V\), modulo weak Schatten ideal \(\mathcal{L}_{p, \infty}\) for arbitrarily small \(p > 0\). As a corollary, we prove a necessary and sufficient condition for the quantised derivative \(i [\operatorname{sgn}(D_m + V), 1 \otimes M_f]\) of a bounded function \(f\) on \(\mathbb{R}^d\) to belong to the weak Schatten ideal \(\mathcal{L}_{d, \infty}\).

MSC:

35Q41 Time-dependent Schrödinger equations and Dirac equations
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
35P15 Estimates of eigenvalues in context of PDEs
45P05 Integral operators
35Q40 PDEs in connection with quantum mechanics
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