The spectrum of all self-adjoint extensions for a general exceptional Laguerre-type differential expressions is given in terms of the Darboux transformations which relate the expression to the classical Laguerre differential expression in the form of an explicit Weyl--Titchmarsh m-function, up to a sign.
The construction relies primarily on boundary triples, which parameterize the self-adjoint extensions and produce the Weyl--Titchmarsh m-functions, and manipulations of Maya diagrams and partitions, which classify the seed functions defining the relevant Darboux transforms.