Proof that the Klein-Gordon type equation with alpha attractor potential has no Liouvillian solution or as a composition of special functions
This study investigates the analytical solvability of the Klein-Gordon and Duffin-Kemmer-Petiau (DKP) equations for a scalar particle interacting with a transcendental $\alpha$-attractor-type potential, $V(x) = V_0 e^{a \tanh(bx)}$. We first address the problem of integrability within the framework of Picard-Vessiot theory. By analyzing the differential field extensions associated with the system, we demonstrated that the differential Galois group is the full special linear group $SL(2, \mathbb{C})$. Given that this group is not solvable, we provide rigorous proof for the non-existence of Liouvillian solutions, effectively ruling out any expression in terms of primitives and elementary functions. Building upon this result, we further establish that wavefunctions cannot be represented as finite compositions or transformations of classical special functions, such as those of the Bessel, Whittaker, or Heun families. This second conclusion is supported by the ``double-transcendence'' of the potential; we prove via the Hermite-Lindemann theorem that no rational coordinate transformation $z(x)$ exists that could map the physical equation into an ordinary differential equations(ODE) with rational coefficients. Consequently, the $\alpha$-attractor potential is strictly non-integrable and lies entirely outside the landscape of solvable relativistic quantum systems.