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In this paper, we develop a systematic approach to treat Dirac operators A η , τ , λ with singular electrostatic, Lorentz scalar, and anomalous magnetic interactions of strengths η , τ , λ ∈ ℝ , respectively, supported on points in ℝ , curves in ℝ 2 , and surfaces in ℝ 3 that is based on boundary triples and their associated Weyl functions. First, we discuss the one-dimensional case which also serves as a motivation for the multidimensional setting. Afterwards, in the two- and three-dimensional situation we construct quasi, generalized, and ordinary boundary triples and their Weyl functions, and provide a detailed characterization of the associated Sobolev spaces, trace theorems, and the mapping properties of integral operators which play an important role in the analysis of A η , τ , λ . We make a substantial step towards more rough interaction supports Σ and consider general compact Lipschitz hypersurfaces. We derive conditions for the interaction strengths such that the operators A η , τ , λ are self-adjoint, obtain a Krein-type resolvent formula, and characterize the essential and discrete spectrum. These conditions include purely Lorentz scalar and purely non-critical anomalous magnetic interactions as well as the confinement case, the latter having an important application in the mathematical description of graphene. Using a certain ordinary boundary triple, we also show the self-adjointness of A η , τ , λ for arbitrary critical combinations of the interaction strengths under the condition that Σ is C ∞ -smooth and derive its spectral properties. In particular, in the critical case, a loss of Sobolev regularity in the operator domain and a possible additional point of the essential spectrum are observed. [ABSTRACT FROM AUTHOR]

We give a self-contained presentation of the theory of self-adjoint extensions using the technique of boundary triples. A description of the spectra of self-adjoint extensions in terms of the corresponding Krein maps (Weyl functions) is given. Applications include quantum graphs, point interactions, hybrid spaces and singular perturbations. [ABSTRACT FROM AUTHOR]

Under a certain condition, we find the explicit formulas for the trace functions of certain intertwining operators among gl(n)-modules, introduced by Etingof in connection with the solutions of the Calogero–Sutherland model. If n = 2, the master function of the trace function is exactly the classical Gauss hypergeometric function. When n > 2, the master functions of the trace functions are a new family of multiple hypergeometric functions, whose differential property and Euler type integral representation are dominated by certain polynomials of integral paths connecting pairs of positive integers. Moreover, we also find the trace functions for sp(2n) explicitly and prove that they give rise to solutions of the Olshanesky–Perelomov model of type C. The master functions of the trace functions for sp(2n) are similar new multiple path hypergeometric functions. Analogous multiple path hypergeometric functions for orthogonal Lie algebras are defined and studied. [ABSTRACT FROM AUTHOR]