Solving the Green's function as a single eigenstate in a boundary value problem

It is well known that the Green’s function of an operator L(x) in a boundary value problem can be expressed as a bilinear expansion using the eigenstates of L(x). Here we introduce an auxiliary eigenvalue problem, from which the Green’s function is uniquely determined by a single eigenstate. This approach is easy to implement numerically, and it becomes very helpful when the eigenstates of L(x) are badly conditioned, for example, when L(x) is non-Hermitian and at an exceptional point. We illustrate this approach in one-dimensional and two-dimensional Helmholtz equations, with a focus on non-Hermitian systems that are due to their openness as well as non-Hermitian potentials.