2012/06/01 by Fumio Hiroshima, Takashi Ichinose, József Lörinczi +1 · 53 citations
Mathematics · Physics and Astronomy · #Fourier integral operator #Fractional Differential Equations Solutions #Laplace operator #Mathematical analysis #Mathematical functions and polynomials #Mathematical physics #Mathematics #Operator theory #Path integral formulation #Physics #Pure mathematics #Quantum #Quantum mechanics #Schrödinger's cat #Semigroup #Spectral Theory in Mathematical Physics #math-ph #math.MP
paper · pdf · doi:10.1142/s0129055x12500134
published in Reviews in Mathematical Physics 24(06), 1250013 (World Scientific) · We revised the first version
openalex publication_date 2012/06/01 · arxiv created 2012/07/09 · arxiv updated 2015/06/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Path integral representations for generalized Schrödinger operators obtained under a class of Bernstein functions of the Laplacian are established. The one-to-one correspondence of Bernstein functions with Lévy subordinators is used, thereby the role of Brownian motion entering the standard Feynman–Kac formula is taken here by subordinate Brownian motion. As specific examples, fractional and relativistic Schrödinger operators with magnetic field and spin are covered. Results on self-adjointness of these operators are obtained under conditions allowing for singular magnetic fields and singular external potentials as well as arbitrary integer and half-integer spin values. This approach also allows to propose a notion of generalized Kato class for which an L p -L q bound of the associated generalized Schrödinger semigroup is shown. As a consequence, diamagnetic and energy comparison inequalities are also derived.