相关论文: On a spectral flow formula for the homological ind…

We develop a principal trace and generalized index formula for a Dirac-Schr\"odinger operator $D$ on open space of odd dimension $d\geq 3$ with a potential given by a family of self-adjoint unbounded operators acting on a infinite…

Let $A(t)$ be a continuous path of Fredhom operators, we first prove that the spectral flow $sf(A(t))$ is cogredient invariant. Based on this property, we give a decomposition formula of spectral flow if the path is invariant under a…

We study the relation between spectral flow and index theory within the framework of (unbounded) KK-theory. In particular, we consider a generalised notion of 'Dirac-Schr\"odinger operators', consisting of a self-adjoint elliptic…

First, we prove a local spectral flow formula (Theorem 3.7) for a differentiable curve of selfadjoint Fredholm operators. This formula enables us to prove in a simple way a general spectral flow formula. Secondly, we prove a splitting…

We consider a continuous curve of self-adjoint Fredholm extensions of a curve of closed symmetric operators with fixed minimal domain $D_m$ and fixed {\it intermediate} domain $D_W$. Our main example is a family of symmetric generalized…

One may trace the idea that spectral flow should be given as the integral of a one form back to the 1974 Vancouver ICM address of I.M. Singer. Our main theorem gives analytic formulae for the spectral flow along a norm differentiable path…

We define a spectral flow for paths of selfadjoint Fredholm operators that are equivariant under the orthogonal action of a compact Lie group as an element of the representation ring of the latter. This $G$-equivariant spectral flow shares…

We construct a Hilbert scale on $L^2([0,1])$ via a unitary twist operator that maps the standard Fourier basis to half-integer frequency exponentials. The resulting weighted spaces, equipped with norms indexed by $(1+|k+\tfrac{1}{2}|^2)^s$,…

If $A \colon D(A) \subset \mathcal{H} \to \mathcal{H}$ is an unbounded Fredholm operator of index $0$ on a Hilbert space $\mathcal{H}$ with a dense domain $D(A)$, then its spectrum is either discrete or the entire complex plane. This…

An analytic definition of a $\mathbb{Z}_2$-valued spectral flow for paths of real skew-adjoint Fredholm operators is given. It counts the parity of the number of changes in the orientation of the eigenfunctions at eigenvalue crossings…

Using the general formalism of [12], a study of index theory for non-Fredholm operators was initiated in [9]. Natural examples arise from $(1+1)$-dimensional differential operators using the model operator $D_A$ in $L^2(\mathbb{R}^2; dt…

Given an essentially unitary contraction and an arbitrary unitary dilation of it, there is a naturally associated spectral flow which is shown to be equal to the index of the operator. This purely operator theoretic result is interpreted in…

We review the concepts of the index of a Fredholm operator, the spectral flow of a curve of self-adjoint Fredholm operators, the Maslov index of a curve of Lagrangian subspaces in symplectic Hilbert space, and the eta invariant of operators…

In this article we give a comprehensive treatment of a `Clifford module flow' along paths in the skew-adjoint Fredholm operators on a real Hilbert space that takes values in KO${}_{*}(\mathbb{R})$ via the Clifford index of…

We relate the spectral flow to the index for paths of selfadjoint Breuer-Fredholm operators affiliated to a semifinite von Neumann algebra, generalizing results of Robbin-Salamon and Pushnitski. Then we prove the vanishing of the von…

The index of a selfadjoint Fredholm operator is zero by the well-known fact that the kernel of a selfadjoint operator is perpendicular to its range. The Fredholm index was generalised to families by Atiyah and J\"anich in the sixties, and…

Let M be an even dimensional compact Riemannian manifold with boundary and let D be a Dirac operator acting on the sections of the Clifford module E over M. We impose certain local elliptic boundary conditions for D obtaining a selfadjoint…

We consider Dirac-Schr\"odinger operators over odd-dimensional Euclidean space. The conditions for the potential are based on those of C. Callias in his famous paper on the corresponding index problem. However, we treat the case where the…

Let~$H_0$ and~$V$ be self-adjoint operators such that~$V$ admits a factorisation $V = F^*JF$ with bounded self-adjoint $J$ and $|H_0|^{1/2}$-compact~$F.$ Flow of singular spectrum of the path of self-adjoint operators $H_0 + rV,$ $r \in…

We define a new topology, weaker than the gap topology, on the space of selfadjoint unbounded operators on a separable Hilbert space. We show that the subspace of selfadjoint Fredholm operators represents the functor $K^1$ from the category…