We show that the operator norm of an arbitrary bivariate polynomial, evaluated on certain spectral projections of spin operators, converges to the maximal value in the semiclassical limit. We contrast this limiting behavior with that of the polynomial when evaluated on random pairs of projections. The discrepancy is a consequence of a type of Slepian spectral concentration phenomenon, which we prove in some cases.
In this paper it is proved that for families of stochastic operators on a countable tensor product, depending smoothly on parameters, any spectral projection persists smoothly, where smoothness is defined using norms based on ideas of Dobrushin. A rigorous perturbation theory for families of stochastic operators with spectral gap is thereby created. It is illustrated by deriving an effective slow 2-state dynamics for a 3-state probabilistic cellular automaton. Some further potential applications are discussed.
The first main result of this note, Theorem 1.2, establishes the determinantal identities (7) and (8) for the expectation, under a determinantal point process governed by an integrable projection kernel, of scaling limits of characteristic polynomials sampled at several points. The determinantal identities (7) and (8) can be seen as the scaling limit of the identity of Fyodorov and Strahov for the averages of ratios of products of the values of the characteristic polynomial of a Gaussian unitary matrix. Borodin, Olshanski and Strahov derived the determinantal identity of Fyodorov and Strahov f…
We consider Fredholm determinants of the form identity minus product of spectral projections corresponding to isolated parts of the spectrum of a pair of self-adjoint operators. We show an identity relating such determinants to an integral over the spectral shift function in the case of a rank-one perturbation. More precisely, we prove $$ -\ln \left(\det \big(\mathbf{1} -\mathbf{1} _{I}(A) \mathbf{1}_{\mathbb R\backslash I}(B)\mathbf{1}_{I}(A)\big) \right) = \int_I \text{d} x \int_{\mathbb R\backslash I} \text{d} y\, \frac{ξ(x)ξ(y)}{(y-x)^2}, $$ where $\mathbf{1}_J (\cdot)$ denotes the spectra…
Spectral asymmetry via pseudodifferential projections: the massless Dirac operator
Università degli Studi di Milano（米兰大学）；Maxwell Institute for Mathematical Sciences, Edinburgh（爱丁堡马克斯韦尔数学科学研究所）；Heriot-Watt University, Edinburgh（爱丁堡赫瑞-瓦特大学）；Università degli Studi di Pavia（帕维亚大学）；INFN, Sezione di Pavia（意大利国家核物理研究所帕维亚分部）；INdAM, Sezione di Pavia（意大利国家高等数学研究所帕维亚分部）
Unique continuation principle for spectral projections of Schr\" odinger operators and optimal Wegner estimates for non-ergodic random Schr\" odinger operators
It is known that the q-Heun equation has polynomial-type solutions in some special cases, and the condition for the accessory parameter E is described by the roots of the spectral polynomial. We investigate the spectral polynomial by considering the ultradiscrete limit.
