2015/09/26 by Palle E. T. Jørgensen, Jorgensen, Palle, Erin P. J. Pearse +3
Mathematics · Physics and Astronomy · #46N30 #58J65 #65R10 #81S25 #FOS: Mathematics #Functional Analysis (math.FA) #Primary 47L60 #Quantum Mechanics and Applications #Quantum optics and atomic interactions #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1509.08024
openalex publication_date 2015/09/26 · openalex created_date 2017/03/03 · openalex updated_date 2026/07/28
Our main theorem is in the generality of the axioms of Hilbert space, and the theory of unbounded operators. Consider two Hilbert spaces such that their intersection contains a fixed vector space D. It is of interest to make a precise linking between such two Hilbert spaces when it is assumed that D is dense in one of the two; but generally not in the other. No relative boundedness is assumed. Nonetheless, under natural assumptions (motivated by potential theory), we prove a theorem where a comparison between the two Hilbert spaces is made via a specific selfadjoint semibounded operator. Applications include physical Hamiltonians, both continuous and discrete (infinite network models), and operator theory of reflection positivity.