On absence of threshold resonances for Schrödinger and Dirac operators

Abstract

Using a unified approach employing a homogeneous Lippmann-Schwinger-type equation satisfied by resonance functions and basic facts on Riesz potentials, we discuss the absence of threshold resonances for Dirac and Schrödinger operators with sufficiently short-range interactions in general space dimensions. More specifically, assuming a sufficient power law decay of potentials, we derive the absence of zero-energy resonances for massless Dirac operators in space dimensions n 3, the absence of resonances at ±m for massive Dirac operators (with mass m > 0) in dimensions n 5, and recall the well-known case of absence of zero-energy resonances for Schrödinger operators in dimension n 5.

The limiting absorption principle for massless Dirac operators, properties of spectral shift functions, and an application to the Witten index of non-Fredholm operators

“…In fact, for n = 3 the absence of zero-energy resonances has been shown under the weaker decay [8]. The absence of zeroenergy resonances for massless Dirac operators in dimensions n 4 as contained in Theorem 10.7 (ii) appears to have gone unnoticed in the literature and was only recently observed in [83]. ⋄…”

“…This is the first result of this kind applicable to non-Fredholm operators in a partial differential operator setting involving multi-dimensional massless Dirac operators. In a sense, a project that started with Pushnitski in 2008, was considerably extended in scope in [77], and further developed with the help of [38]- [44], [82], [83], finally comes full circle.…”

Limiting absorption principle and virtual levels of operators in Banach spaces

“…Let us mention the dichotomy between a virtual level and an eigenvalue manifested in the large-time behavior of the heat kernel and the behavior of the Green function near criticality; see [Pin92,Pin04]. We also mention recent articles [BBV20] on properties of virtual states of selfadjoint Schrödinger operators and [GN20] proving the absence of genuine (non-L 2 ) virtual states of selfadjoint Schrödinger operators and massive and massless Dirac operators, as well as giving classification of virtual levels and deriving properties of eigenstates and virtual states.…”

“…For more details and references, see [BC21]. Related results on properties of eigenstates and virtual states are in [GN20] (Schrödinger and massive Dirac operators in dimension d ≥ 3 and massless Dirac operators in d ≥ 2) and in [BBV20, Theorem 2.3] (Schrödinger operators in d ≤ 2). Let us note that, prior to [BC21], the nonselfadjoint case has not been considered (although some results appeared in [CP05]).…”

The Limiting Absorption Principle for Massless Dirac Operators, Properties of Spectral Shift Functions, and an Application to the Witten Index of Non-Fredholm Operators

“…I H 2 ; H 1 /, denoted by L .0 C I H 2 ; H 1 /, and L .0 C I H This is the first result of this kind applicable to non-Fredholm operators in a partial differential operator setting involving multi-dimensional massless Dirac operators. In a sense, a project that started with Pushnitski in 2008, was considerably extended in scope in [78], and further developed with the help of [38,[41][42][43][44]82,83], finally comes full circle.…”