arXiv:2603.01709v2 Announce Type: replace Abstract: In this work, we study long-time numerical integration of Hamiltonian systems subject to linear perturbations. We assume the energy-compatible linear-damping condition, which ensures that the physical energy is nonincreasing along the linear damping subflow. Since the conservative and dissipative effects cannot always be merged into a single gradient-structured dissipation and classical energy-stable methods developed for gradient flows cannot directly extend to this setting, we propose a unified framework of two efficient and energy-stable splitting exponential integrators (SEI) to separately handle the dissipative and conservative parts: SEISAV (SEI based on the scalar auxiliary variable) and SEILM (SEI based on Lagrange multiplier). The SEISAV scheme composes the exact damping subflow with an exponential integrator based on the SAV update for the Hamiltonian subflow and requires solving only a scalar linear equation at each time step. We prove a discrete modified-energy dissipation law for SEISAV that mirrors the continuous energy-dissipation mechanism. To enforce decay of the original energy rather than a modified one, we further develop SEILM by incorporating a Lagrange-multiplier formulation within the splitting exponential framework, leading to solving only a scalar nonlinear equation at each time step. Numerical experiments on representative linearly damped Hamiltonian models confirm the designed second-order accuracy, energy stability, and competitive efficiency relative to state-of-the-art schemes, indicating that the proposed framework provides a simple and robust approach for simulating linearly perturbed Hamiltonian dynamics.