Tunneling dynamics in relativistic and n

This study investigates the tunneling dynamics in relativistic and nonrelativistic wave equations, focusing on solutions for a source with a sharp onset and frequencies below cut-off. We derive the solutions for both types of equations and demonstrate that a scaling of position and time simplifies nonrelativistic solutions to a single case, but not for the relativistic equation, leading to the identification of qualitatively distinct 'shallow' and 'deep' tunneling regimes. We find that nonrelativistic wave equations do not display tunneling behavior beyond a specific penetration length, whereas the corresponding relativistic forerunner exhibits complex oscillatory behavior due to interference from two saddle point contributions, along with two distinct arrival times for maxima in amplitude. Furthermore, we identify an early relativistic forerunner that does exhibit tunneling, highlighting that within the penetration length, tunneling is significantly more pronounced for precursors of the relativistic equation. The findings contribute to a deeper understanding of tunneling in various wave equation frameworks.