PSR J0205+6449 is a young rotation-powered pulsar in SNR 3C 58. It is one of only three young (<10,000 year old) pulsars which are so far detected in the radio and the classical X-ray bands, as well as at hard X-rays above 20 keV and at high-energy (>100 MeV) $gamma$-rays. The other two young pulsars are the Crab and PSR B1509-58. Our aim is to derive the timing and spectral characteristics of PSR J0205+6449 over the broad X-ray band from ~0.5 to ~270 keV. We used all publicly available RXTE observations of PSR J0205+6449 to first generate accurate ephemerides over the period September 30, 2000 - March 18, 2006. Next, phase-folding procedures yielded pulse profiles using data from RXTE PCA and HEXTE, and XMM-Newton EPIC PN. While our timing solutions are consistent with earlier results, our work shows sharper structures in the PCA X-ray profile. The X-ray pulse profile consists of two sharp pulses, separated in phase by 0.488(2), which can be described with 2 asymmetric Lorentzians, each with the rising wing steeper than the trailing wing, and full-width-half-maximum 1.41(5) ms and 2.35(22) ms, respectively. We find an indication for a flux increase by a factor ~2, about 3.5 sigma above the time-averaged value, for the second, weaker pulse during a two-week interval, while its pulse shape did not change. The spectrum of the pulsed X-ray emission is of non-thermal origin, exhibiting a power-law shape with photon index Gamma = 1.03(2) over the energy band ~0.5 to ~270 keV. In the energy band covered with the PCA (~3-30 keV) the spectra of the two pulses have the same photon index, namely, 1.04(3) and 1.10(8), respectively.
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