A remark on a paper of E. B. Davies Zworski, Maciej
Proceedings of the American Mathematical Society,
10/2001, Letnik:
129, Številka:
10
Journal Article
Recenzirano
Odprti dostop
We explain the existence of open sets of complex quasi-modes in terms of Hörmander's commutator condition.
The Lie-Trotter formula, together with its higher-order generalizations, provides a direct approach to decomposing the exponential of a sum of operators. Despite significant effort, the error scaling ...of such product formulas remains poorly understood. We develop a theory of Trotter error that overcomes the limitations of prior approaches based on truncating the Baker-Campbell-Hausdorff expansion. Our analysis directly exploits the commutativity of operator summands, producing tighter error bounds for both real- and imaginary-time evolutions. Whereas previous work achieves similar goals for systems with geometric locality or Lie-algebraic structure, our approach holds, in general. We give a host of improved algorithms for digital quantum simulation and quantum Monte Carlo methods, including simulations of second-quantized plane-wave electronic structure,k-local Hamiltonians, rapidly decaying power-law interactions, clustered Hamiltonians, the transverse field Ising model, and quantum ferromagnets, nearly matching or even outperforming the best previous results. We obtain further speedups using the fact that product formulas can preserve the locality of the simulated system. Specifically, we show that local observables can be simulated with complexity independent of the system size for power-law interacting systems, which implies a Lieb-Robinson bound as a by-product. Our analysis reproduces known tight bounds for first- and second-order formulas. Our higher-order bound overestimates the complexity of simulating a one-dimensional Heisenberg model with an even-odd ordering of terms by only a factor of 5, and it is close to tight for power-law interactions and other orderings of terms. This result suggests that our theory can accurately characterize Trotter error in terms of both asymptotic scaling and constant prefactor.
This article deals with the stability of discrete-time switched linear systems whose all subsystems are unstable and the set of admissible switching signals obeys prespecified restrictions on ...switches between the subsystems and dwell times on the subsystems. We derive sufficient conditions on the subsystems matrices such that a switched system is globally exponentially stable under a set of purely time-dependent switching signals that obeys the given restrictions. The main apparatuses for our analysis are (matrix) commutation relations between certain products of the subsystems matrices and graph-theoretic arguments.
The growth of commutators of initially commuting local operators diagnoses the onset of chaos in quantum many-body systems. We compute such commutators of local field operators with N components in ...the (2+1)-dimensional O(N) nonlinear sigma model to leading order in 1/N. The system is taken to be in thermal equilibrium at a temperature T above the zero temperature quantum critical point separating the symmetry broken and unbroken phases. The commutator grows exponentially in time with a rate denoted λL. At large N the growth of chaos as measured by λL is slow because the model is weakly interacting, and we find λL≈3.2T/N. The scaling with temperature is dictated by conformal invariance of the underlying quantum critical point. We also show that operators grow ballistically in space with a “butterfly velocity” given by vB/c≈1 where c is the Lorentz-invariant speed of particle excitations in the system. We briefly comment on the behavior of λL and vB in the neighboring symmetry broken and unbroken phases.
The spectrum of two-dimensional adjoint QCD is surprisingly insensitive to the number of colors \(N_c\) of its gauge group. It is argued that the cancellation of finite \(N_c\) terms is rather ...natural and a consequence of the singularity structure of the theory. In short, there are no finite \(N_c\) contraction terms, hence there cannot be any finite \(N_c\) singular terms, since the former are necessary to guarantee well-behaved principal value integrals. We evaluate and categorize the matrix elements of the theory's light-cone Hamiltonian to show how terms emerging from finite \(N_c\) contributions to the anti-commutator cancel against contributions from the purely finite \(N_c\) term of the Hamiltonian. The cancellation is not complete; finite terms survive and modify the spectrum, as is known from numerical work.
Wear behaviour determines the service life and reliability of the motor’s commutator and Brush, including mechanical and electrical wear. The temperature and relative humidity, especially humid ...tropical climate conditions, significantly affect the wear behaviour of commutator brushes. The traditional life tests are often manual and time-consuming due to a long expected life. This paper presents study results on the service life and reliability of graphite brushes when the temperature, relative humidity, and current change. The calculation method of service life and reliability depending on wear testing was used to evaluate the service life characteristics. The experimental results, service life, and reliability calculations showed that the service life and reliability of the graphite brushes were influenced clearly by current, temperature, relative humidity, and the probability of failure-free operation. It depends on the current more than relative humidity and temperature. In the range of changing current in this study, the service life: TP(99%) ~ 0.54*TP(50%), TP(95%) ~ 0.62*TP(50%), and TP(90%) ~ 0.68*TP(50%).
In this paper, the authors consider the endpoint estimates for the maximal Calderón commutator defined by ...$$T_{\Omega,\,a}^*f(x)=\sup_{\epsilon>0}\Big|\int_{|x-y|>\epsilon}\frac{\Omega(x-y)}{|x-y|^{d+1}} \big(a(x)-a(y)\big)f(y)dy\Big|,$$ where \(\Omega\) is homogeneous of degree zero, integrable on \(S^{d-1}\) and has vanishing moment of order one, \(a\) be a function on \(\mathbb{R}^d\) such that \(\nabla a\in L^{\infty}(\mathbb{R}^d)\). The authors prove that if \(\Omega\in L\log L(S^{d-1})\), then \(T^*_{\Omega,\,a}\) satisfies an endpoint estimate of \(L\log\log L\) type.
We study the trace ideal properties of the commutators (−Δ)ϵ2,Mf of a power of the Laplacian with the multiplication operator by a function f on Rd. For a certain range of ϵ∈R, we show that this ...commutator belongs to the weak Schatten class Ld1−ϵ,∞ if and only if the distributional gradient of f belongs to Ld1−ϵ. Moreover, in this case we determine the asymptotics of the singular values. Our proofs use, among other things, the tool of Double Operator Integrals.
We discuss the counting of Nambu-Goldstone (NG) modes associated with the spontaneous breaking of higher-form global symmetries. Effective field theories of NG modes are developed based on ...symmetry-breaking patterns, using a generalized coset construction for higher-form symmetries. We derive a formula of the number of gapless NG modes, which involves expectation values of the commutators of conserved charges, possibly of different degrees.
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