On the spectral decomposition of the complex Robin Laplacian

The mathematical properties of the Laplacian on a bounded domain are well-known when the boundary condition is of the first type (Dirichlet) or second type (Neumann). In both cases, this operator is self-adjoint and, therefore, diagonalizable, its spectrum is discrete, and the set of eigenfunctions can be chosen to form an orthonormal basis of the Hilbert space of square-integrable functions on the domain. However, in the case of the third type (Robin) boundary condition, the same is true only when the parameter is real-valued. On the contrary, when this parameter is complex-valued, the Laplacian may not even be diagonalizable. In this paper, the spectral decomposition of the complex Robin Laplacian is investigated in the most general case possible, and a formula that decomposes any square-integrable function on the set of its (generalized) eigenfunctions is provided. This result is applied to the Green's function of the Helmholtz equation, whose existence, unicity, and closed-form expression are established in this general setting, and the statistical wave field theory, which provides the statistical laws of waves propagating in a bounded domain.