Path Integral Representation for Schrödinger Operators With Bernstein Functions of the Laplacian

Abstract

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.

Absence of energy level crossing for the ground state energy of the Rabi model

“…We can conjecture that the ground state energy of the Rabi model has no energy level crossing. In this paper we prove this fact with the functional-integral method [12,11] as a corollary of the fact stating that the ground state energy of the Rabi model is simple (i.e., the ground state is unique). It reveals us that it is in the ultra-strong coupling regime of circuit QED that there is a big qualitative difference as well as quantitative one between the Rabi model and the JC model.…”

“…We will construct a path integral representation of e −tHR to show that the ground state energy is simple. This is a minor modification of recent papers [12,11], where the Feynman-Kac type formula with spin is established. In particular the spin-boson model is studied by path measure in [11] and we can apply it in this paper since the Rabi model can be regarded as the single mode photon version of the spin-boson model.…”

The spectrum of non-local discrete Schrödinger operators with a δ-potential

“…In [14] we have introduced a class of generalized Schrödinger operators whose kinetic term is given by so called Bernstein functions of the Laplacian. These operators are non-local and via a Feynman-Kac representation generate subordinate Brownian motion killed at a rate given by the potential.…”

On Three Magnetic Relativistic Schrödinger Operators and Imaginary-Time Path Integrals

“…Here n(dy) = n(y)dy is an m-dependent measure on R d \ {0}, called Lévy measure with density 5) and appears in the Lévy-Khinchin formula:…”