What question did this study set out to answer?

This study aims to develop a generalized index formula for Dirac–Schrödinger operators with unbounded potentials in odd dimensions.

June 27, 2026Open Access

Trace and index of Dirac–Schrödinger operators on open space with operator potentials

Puntos clave

This study aims to develop a generalized index formula for Dirac–Schrödinger operators with unbounded potentials in odd dimensions.
Developed a principal trace and index formula for Dirac–Schrödinger operators in odd dimensions d≥3.
Generalized the Callias index theorem to include unbounded operator potentials.
Provided examples calculating the Witten index for non-Fredholm massless Dirac–Schrödinger operators.
Introduced a novel trace formula applicable for non-Fredholm scenarios.
Demonstrated that the generalized index formula can assess operators that are not Fredholm.
Calculated the Witten index using the trace formula.

Resumen

We develop a principal trace and generalized index formula for a Dirac–Schrödinger operator D on open space of odd dimension d 3 with a potential given by a family of self-adjoint unbounded operators acting on an infinite-dimensional Hilbert space H. The presented results generalize formulas surrounding the Callias index theorem to the case of unbounded operator potentials, for which the operator D is not necessarily Fredholm. This is the principal novelty of this paper. As application, we include examples where the trace formula is used to calculate the Witten index of non-Fredholm massless (d+1) -Dirac–Schrödinger operators acting in L^2 (R^d+1, H).

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