Quantum time delay for unitary operators: General theory

We present a suitable framework for the definition of quantum time delay in terms of sojourn times for unitary operators in a two-Hilbert spaces setting. We prove that this time delay defined in terms of sojourn times (time-dependent definition) exists and coincides with the expectation value of a unitary analogue of the Eisenbud–Wigner time delay operator (time-independent definition). Our proofs rely on a new summation formula relating localization operators to time operators and on various tools from functional analysis such as Mackey’s imprimitivity theorem, Trotter–Kato formula and commutator methods for unitary operators. Our approach is general and model-independent.

Hölder Continuous Solutions of Degenerate Differential Equations with Finite Delay

Using known operator-valued Fourier multiplier results on vector-valued Hölder continuous function spaces Cα(ℝ; X), we completely characterize the Cα-well-posedness of the first order degenerate differential equations with finite delay (Mu)′(t) = Au(t) + Fut + f(t) for t ∊ ℝ by the boundedness of the (M, F)-resolvent of A under suitable assumption on the delay operator F, where A, M are closed linear operators on a Banach space X satisfying D(A) İ D(M) ≠ = ﹛0﹜, the delay operator F is a bounded linear operator from C([−r, 0]; X) to X, and r > 0 is fixed.

An Improved TA Suppression Method for Coded PR Channels

Thermal asperity (TA) resulting from the collision between the slider and the asperity on a magnetic medium during read process can deteriorate the performance of hard disk drives (HDDs). Without TA detection and correction algorithms, the system performance can be unacceptable, depending on how severe the TA is. This paper presents an improved TA suppression method for coded partial response (PR) channels, which consists of two channels running in parallel. Specifically, one channel is matched to the target H(D), while the other is matched to the target H(D)G(D), where G(D) = 1 – D2 is a bandpass filter and D is a delay operator. The soft-output Viterbi algorithm (SOVA) detector in the H(D) channel yields the high-quality soft information in absence of the TA, while that in the G(D)H(D) channel produces the high-quality soft information in presence of the TA. Then, the overall soft information chosen from these two detectors, depending on if a TA is detected or not, is sent to the decoder according to the turbo equalization principle. Experimental results show that the proposed method performs better than the conventional and the previously proposed ones, when operating at high signal-to-noise ratio (SNR) region where a practical HDD works.

Dichotomy and positivity of neutral equations with nonautonomous past

Consider the linear partial neutral functional differential equations with nonautonomous past of the form (?/?t) F(u(t, ?)) = BFu(t, ?) + ?u(t, ?), t ? 0; (? / ?t) u(t, s) = (? / ?s) u(t, s) + A(s)u(t, s), t ? 0 ? s, where the function u(?, ?) takes values in a Banach space X. Under appropriate conditions on the difference operator F and the delay operator ? we prove that the solution semigroup for this system of equations is hyperbolic (or admits an exponential dichotomy) provided that the backward evolution family U = (U(t, s))t?s?0 generated by A(s) is uniformly exponentially stable and the operator B generates a hyperbolic semigroup (etB)t?0 on X. Furthermore, under the positivity conditions on (etB)t?0, U, F and ? we prove that the above-mentioned solution semigroup is positive and then show a sufficient condition for the exponential stability of this solution semigroup.