Slice hyperholomorphicity of the -resolvent operators and boundary conditions
arXiv:2602.04424
Abstract
The foundation of spectral theory on the -spectrum can be traced back to the quaternionic framework of quantum mechanics. The concept of -spectrum for quaternionic operators emerged as the natural spectrum in slice hyperholomorphic functional calculi, known as the -functional calculus and also utilized in the quaternionic spectral theorem. This spectral theory extends to Clifford operators. A key distinction from classical complex spectral theory lies in the definition of the -spectrum, which is second order in the operator , and in the -resolvent operators that turns out to be the product of two different operators. This study delves into the analyticity of the -resolvent operators under specified boundary conditions for the -spectral problem. The spectral theory on the -spectrum also provides deeper insights into classical spectral theory.