Exponential growth and decay estimates for eigenfunctions with complex eigenvalues

We prove a priori exponential growth and decay estimates for locally square integrable eigenfunctions with complex eigenvalues of the non-self-adjoint Schr\"odinger operator. There appear two natural critical exponents, given by the imaginary parts of square roots of twice an eigenvalue. Any associated eigenfunction has the same exponential growth or decay rate as one of these, or greater than the upper one. The results may be seen as extensions of the Phragm\'en--Lindel\"of theorem, the Agmon estimates and Rellich's theorem to complex eigenvalues. For the proofs we adapt a commutator scheme of Ito--Skibsted~(2020) to the non-self-adjoint framework, presenting a unified approach to all of them. In order to control contributions from the imaginary parts of a potential and an eigenvalue, an additional commutator estimate is employed.