  A pulsed electron lens produces a betatron tune shift along a hadron bunch as
a function of the longitudinal coordinates, which is a longitudinal detuning.
An example of transverse detuning are the tune shifts due to octupole magnets.
This paper considers a pulsed electron lens as a measure to mitigate the
head-tail instabilities. Using a detailed analytical description within a
Vlasov formalism, the coherent properties of the longitudinal and transverse
detuning are presented. The analytical predictions are compared with the
results of the particle tracking simulations. A pulsed electron lens is
demonstrated to be a source of tune spread with two components: a static one,
leading to Landau damping; and a dynamic one, leading to an effective impedance
modification, which we demonstrate analytically and in our particle tracking
simulations. The effective impedance modification can be important for beam
stability due to devices causing a longitudinal detuning, especially for
nonzero head-tail modes. The Vlasov formalism is extended to include the
combination of longitudinal and transverse detuning. As a possible application
at the SIS100 heavy-ion synchrotron (FAIR at GSI Darmstadt, Germany), a
combination of a pulsed electron lens with octupole magnets is considered.
