Int J Theor Phys (2012) 51:526–535 DOI 10.1007/s10773-011-0931-7 Relativistic Bosons in Time-Harmonic Electric Fields Ovidiu Buhucianu · Marina-Aura Dariescu · Ciprian Dariescu Received: 15 June 2011 / Accepted: 17 August 2011 / Published online: 10 September 2011 © Springer Science+Business Media, LLC 2011 Abstract In the present paper, we consider a bi-dimensional thin sample, placed in a strong harmonically oscillating electric field and a static magnetic induction, both directed along the normal to the sample’s plane. The Klein–Gordon equation describing the relativistic bosons leads to a Mathieu’s type equation for the temporal part of the wave functions. It follows that, for the electric field pulsation inside a computable range, depending on the external fields intensities, the amplitude functions are turning from oscillatory to exponen- tially growing modes. For ultra-relativistic particles, one can recover the periodic stationary amplitude behavior. Keywords Klein–Gordon equation · Parabolic cylinder functions · Mathieu’s equation · Mathieu’s functions 1 Introduction In spite of the fact that the substance is made of fermions and the spinless bosons are not easily available in nature, the study of these charged particles moving in different external fields is of interest since, to a very good degree of approximation, one can find measurable properties of mesoscopic systems that do actually behave as it is predicted by such theoreti- cal models. For periodic fields, one usually has to deal with the celebrated Mathieu’s equation, which have been formulated more than one century ago, when Mathieu investigated the vibrations of an elliptical drum [1, 2]. Among the first important results coming from this linear second-order differential equation, with applications in physics and engineering, we mention the one belonging to McLachlan, who has found a method of producing a frequency modulated wave from an audio signal generated by a capacitive microphone, tuned to the oscillating open circuit [3]. O. Buhucianu · M.-A. Dariescu ( ) · C. Dariescu Faculty of Physics, Alexandru Ioan Cuza University of Iasi, Bd. Carol I no. 11, 700506 Iasi, Romania e-mail: marina@uaic.ro