The double angle theorems of Davis and Kahan bound the change in an invariant subspace when a Hermitian matrix A is subject to an additive perturbation A → Ã = A+1A. This paper supplies analogous results when A is subject to a congruential, or multiplicative, perturbation A → Ã = D∗AD. The relative gaps that appear in the bounds involve the spectrum of only one matrix, either A or Ã, in contras...

Two toy Hamiltonians of anharmonic oscillators are considered. Both of them are found exactly solvable not only in the limit of infinite spatial dimension D but also in the next D ≫ 1 approximation. This makes all their bound states tractable via perturbation series in 1/ √ D ≪ 1.

Recently developed time-independent bound-state perturbation theory is extended to treat the scattering domain.The changes in the partial wave phase shifts are derived explicitly and the results are compared with those of other methods.

This paper studies the Schatten-q error of low-rank matrix estimation by singular value decomposition under perturbation. We specifically establish a perturbation bound on via projection bound. Then, we lower bounds to justify tightness upper error. further develop user-friendly sin? for subspace based Finally, demonstrate advantage our results over ones in literature simulation.

As increasing amounts of sensitive personal information finds its way into data repositories, it is important to develop analysis mechanisms that can derive aggregate information from these repositories without revealing information about individual data instances. Though the differential privacy model provides a framework to analyze such mechanisms for databases belonging to a single party, th...

Recent perturbation results for exponentially dichotomous operators are generalized, in part by replacing compactness conditions on the perturbation by resolvent compactness. Both additive and multiplicative perturbations are considered. Mathematics Subject Classification (2000). Primary 47D06; Secondary 47A55.