9 problems

Koplienko's conjecture. There exists a real Borel measure $\nu_p=\nu_{p,H_0,V}$ whose total variation satisfies

Let $\mathcal H$ be a separable complex Hilbert space, let $A$ be a self-adjoint, not necessarily bounded, operator on $\mathcal H$, let $B\in\mathcal S^1(\mathcal H)$ be self-adjo…

Averaged local single-site SSF conjecture. In this setting, the averaged SSF should belong to

Let $H_0$ and $H_1$ be the operators in the scattering framework, and let $V=H_1-H_0$ be a perturbation for which the usual spectral shift function need not exist. At a real number…

Let $\mathcal H$ be a Hilbert space, let $U\in B(\mathcal H)$ be a unitary operator, and let $Z\in \mathcal S^1(\mathcal H)$ be self-adjoint. For a function $f$ on the unit circle…

Let $H_r=H_0+rV$ be a path of self-adjoint operators with trace-class perturbation $V$, and suppose that the absolutely continuous spectrum is the same for all operators $H_r$ in t…

Let $\xi_L(E)$ be the finite-volume spectral shift function for the pair of self-adjoint operators $(-\Delta_L/2+V,-\Delta_L/2)$ on $\mathrm{L}^2(\Lambda_L)$ with Dirichlet boundar…

Let $H_0$ be a self-adjoint operator and let $V$ be a trace-class self-adjoint operator on a Hilbert space. Suppose that the pair $(H_0,V)$ is irreducible, meaning that there is no…

Let $H_0$ and $H_1=H_0+V$ be self-adjoint operators with trace-class self-adjoint perturbation $V$, and let $\xi^{(s)}_{H_1,H_0}$ denote the singular part of the spectral shift fun…