Analytic spectral flow formula for unitaries and Levinson's theorem

Abstract: We prove an integral formula for the spectral flow of differentiable loops of unitaries of the form ${\rm Id}+$Schatten. Our formula is in terms of a regularised winding number, expressed in terms of exact differential forms, and we show how the formula extends to non-closed paths. Applying these ideas to the scattering operator of Schrödinger scattering systems yields explicit formulae for the number of bound states, possibly modified by the presence of resonances, of the system in terms of the potential. We finish by briefly considering the paths of unbounded operators obtained from unitary loops via the Cayley transform. These include cases of moving domain as well as paths with non-constant Hilbert space.

• The paper introduces an analytic spectral flow formula for Schatten-class perturbations of the identity, extending traditional trace-class methods.
• It provides explicit, corrected integral formulae for spectral flow and applies them to unify both regular and resonant cases in Levinson’s theorem for Schrödinger operators.
• The methods extend to non-densely defined operators and link spectral flow with Kasparov K-theory, offering new insights into quantum scattering and index theory.

Analytic Spectral Flow Formula for Unitaries and Its Applications to Levinson’s Theorem

Introduction and Main Contributions

The paper "Analytic spectral flow formula for unitaries and Levinson's theorem" (2604.22451) develops a comprehensive analytic framework for spectral flow along paths of unitary operators that are Schatten-class perturbations of the identity. It establishes general integral formulae, expressed in terms of regularized winding numbers using exact differential forms, supplants the classical trace-class-based approaches, and extends spectral flow computation to non-closed paths. Applying these results to systems in quantum scattering theory, the authors give explicit, corrected forms of Levinson’s theorem for Schrödinger operators of arbitrary dimension, unifying both regular and resonant cases. The analytic methods are further generalized to spectral flow for paths of unbounded operators, translating the problem into the context of Kasparov K-theory and providing new insight into situations with moving domains and Hilbert spaces.

Spectral Flow for Schatten-Class Unitaries: Integral Formulae

One of the significant advances in this paper is the derivation of explicit, analytic integral formulae for the spectral flow along differentiable loops of unitaries of the form ${\rm Id}+$Schatten. Traditionally, spectral flow in the unitary setting is associated with trace-class perturbations, using the well-known formula: $$\text{sf}(U_\bullet) = \frac{1}{2\pi i}\int_0^1 \text{Tr}(U_t^\ast \dot{U}_t) dt$$ This, however, fails for higher ($p>1$) Schatten ideals, particularly when the trace integral diverges or is otherwise ill-defined, as is the case in quantum scattering at high energies or in higher dimensions.

The core results (Theorem \ref{thm:cayley-spec-flow-formula} and Corollary \ref{cor:cayley-spec-flow-formula} in the paper) establish that for a path $(U_t)$ of differentiable $p$-Schatten unitaries, spectral flow can be computed by: $$\text{sf}(U_\bullet) = \frac{-i}{2^{2r+1}} \frac{\Gamma(r+1)}{\sqrt{\pi}\Gamma(r+1/2)} \int_0^1 \text{Tr}\left(U_t^* \dot{U}_t |U_t-\mathbb{I}|^{2r}\right) dt$$ for $r\ge (p-1)/2$, or alternatively,

$$\text{sf}(U_\bullet) = (-1)^n \frac{1}{2\pi i} \int_0^1 \text{Tr}\left(U_t^* \dot{U}_t (U_t-\mathbb{I})^{n}\right) dt$$

for $n\ge p-1$. Here, the integral is of an exact one-form over the Banach-Lie group of unitaries of the form ${\rm Id}+$Schatten, guaranteeing well-definedness and independence of representatives.

A crucial feature is the extension to non-closed paths by choosing canonical geodesic connections at endpoints, with correction terms that generalize the eta invariant in the self-adjoint setting. This is vital for applications to scattering theory, where the spectral parameter runs over $$\text{sf}(U_\bullet) = \frac{1}{2\pi i}\int_0^1 \text{Tr}(U_t^\ast \dot{U}_t) dt$$0.

Regularized Determinant Formulation and Exactness of One-Forms

The paper presents a determinant-based formula connecting the regularized (Schatten-class) determinant $$\text{sf}(U_\bullet) = \frac{1}{2\pi i}\int_0^1 \text{Tr}(U_t^\ast \dot{U}_t) dt$$1 of a path of unitaries to its spectral flow: $$\text{sf}(U_\bullet) = \frac{1}{2\pi i}\int_0^1 \text{Tr}(U_t^\ast \dot{U}_t) dt$$2 This result (Theorem \ref{thm:spec-flow-formula-det}) is a manifestation of the regularized winding number in infinite dimensions and is derived by careful analysis of differential forms on Banach-Lie groups and the analytic properties of regularized determinants.

Moreover, two families of one-forms ($$\text{sf}(U_\bullet) = \frac{1}{2\pi i}\int_0^1 \text{Tr}(U_t^\ast \dot{U}_t) dt$$3 and $$\text{sf}(U_\bullet) = \frac{1}{2\pi i}\int_0^1 \text{Tr}(U_t^\ast \dot{U}_t) dt$$4) are shown to be exact on $$\text{sf}(U_\bullet) = \frac{1}{2\pi i}\int_0^1 \text{Tr}(U_t^\ast \dot{U}_t) dt$$5, which is central to establishing that the corresponding integrands yield homotopy-invariant and additive spectral flow functionals. The normalization is explicitly checked, which guarantees integer-valuedness and connects the analytic to the topological definition via degree theory for the determinant.

Application: Levinson's Theorem in Multi-Dimensional Scattering

A pivotal application is in the analytic formulation of Levinson’s theorem. In quantum scattering, Levinson's theorem relates the total number of bound states to phase shifts or to the winding of the determinant of the scattering matrix as a function of energy. However, in higher dimensions or with long-range/complicated potentials, classical winding number integrals diverge.

The regularized spectral flow formula is applied to the family $$\text{sf}(U_\bullet) = \frac{1}{2\pi i}\int_0^1 \text{Tr}(U_t^\ast \dot{U}_t) dt$$6 of scattering matrices for the perturbed Schrödinger operator $$\text{sf}(U_\bullet) = \frac{1}{2\pi i}\int_0^1 \text{Tr}(U_t^\ast \dot{U}_t) dt$$7: $$\text{sf}(U_\bullet) = \frac{1}{2\pi i}\int_0^1 \text{Tr}(U_t^\ast \dot{U}_t) dt$$8 where $$\text{sf}(U_\bullet) = \frac{1}{2\pi i}\int_0^1 \text{Tr}(U_t^\ast \dot{U}_t) dt$$9 and $p>1$0 are explicit polynomials encoding counterterms derived from heat kernel expansions and high-energy asymptotics. The paper rigorously shows that this corrects for divergence and yields:

• The total number of bound states (including possible resonance corrections in dimensions $p>1$1).
• A unified treatment of both regular and resonant cases, with explicit correction terms $p>1$2 characterizing resonance contribution.

Notably, the formula is dimension-independent and accommodates the presence of threshold resonances, fully clarifying the analytic content of Levinson’s theorem across all physically relevant scenarios.

Spectral Flow for Paths of Non-Densely Defined and Varying Hilbert Space Operators

By leveraging the Cayley transform, the authors generalize spectral flow to paths of self-adjoint operators with possibly moving domains or acting on varying Hilbert spaces, which naturally arise in boundary value problems and in the context of operator $p>1$3-theory. The paper defines a class $p>1$4 of “Schatten–Fredholm” operators (possibly unbounded, possibly non-dense domain), shows that the Cayley transform is a homeomorphism to Schatten-perturbed unitaries, and characterizes continuity via graph projections and Schatten norm topology.

This extension is theoretically potent: it allows for the definition and calculation of spectral flow even in situations where traditional gap or norm-resolvent continuity fails, or where the underlying Hilbert space (domain) itself varies. The theoretical apparatus is then transported into the framework of Kasparov’s $p>1$5-theory, characterizing classes in $p>1$6 via continuous fields of self-adjoint operators, and elucidating the links between analytical and topological invariants.

Theoretical and Practical Implications

The analytic machinery developed here has strong implications:

• Scattering Theory: Provides general formulas for counting bound states and resonance corrections in multidimensional and/or non-compact quantum systems, settling ambiguities in previous treatments of Levinson’s theorem.
• Noncommutative Geometry and $p>1$7-theory: The understanding of spectral flow for generalized paths, including those with moving domains, opens new avenues for index pairings, unbounded $p>1$8-theory, and noncommutative geometry, especially in the study of boundary value problems and spectral invariants.
• Mathematical Physics: The explicit analytic corrections obtained are directly applicable to spectral and scattering problems where the classical theory is intractable due to divergences or topological subtleties.

Future directions include extending these techniques to even broader classes of perturbations, refining analytic torsion or eta-invariants in infinite dimensions, and leveraging the determinant formulas for quantum field theoretic calculations or in the study of anomalies.

Conclusion

The paper provides a rigorous analytic theory of spectral flow for paths of Schatten-class unitary operators, superseding previous approaches limited to trace class, and grounds the application of spectral flow in Levinson’s theorem for Schrödinger operators in any dimension. Precise connections to regularized determinants and graph projection continuity are developed, making the machinery robust for both theoretical development and concrete applications in mathematical physics and $p>1$9-theory. The work not only clarifies the topological-analytic correspondences but also provides practical formulae for spectral invariants in scattering and geometry.