## A probabilistic principle and generalized Schrödinger perturbation.(English)Zbl 0748.60069

Summary: We present a probabilistic principle: for an additive functional of certain Markov processes, its smallness with time is equivalent to its smallness with space, and apply it to a perturbation problem for relativistic Schrödinger operators.

### MSC:

 60J55 Local time and additive functionals 60H25 Random operators and equations (aspects of stochastic analysis) 81Q15 Perturbation theories for operators and differential equations in quantum theory
Full Text:

### References:

 [1] Aizenman, N.; Simon, B.: Brownian motion and Harnack inequality for Schrödinger operators. Comm. pure appl. Math. 35, 209-273 (1982) · Zbl 0459.60069 [2] Blumenthal, R. M.; Getoor, R. K.: Markov processes and potential theory. (1968) · Zbl 0169.49204 [3] Carmona, R.; Masters, W. C.; Simon, B.: Relativistic Schrödinger operators: asymptotic behavior of the eigenfunctions. J. funct. Anal. 91, 117-142 (1990) · Zbl 0716.35006 [4] Chung, K. L.: Lecture from Markov processes to Brownian motion. (1982) · Zbl 0503.60073 [5] Conlon, J.: The ground state energy of a classical gas. Comm. math. Phys. 94, 439 (1984) · Zbl 0946.82501 [6] Fefferman, C.: The N-body problem in quantum mechanics. Comm. pure appl. Math. 39, S67-S109 (1986) · Zbl 0623.46041 [7] Herbst, I. W.: Spectral theory of the operator (p2 + m2) 1 2 - ze 2 r. Comm. math. Phys. 53, 285-294 (1977) · Zbl 0375.35047 [8] Herbst, I. W.; Sloan, A. D.: Perturbation of translation invariant positivity preserving semigroups on $$L2(RN)$$. Trans. amer. Math. soc. 236, 325-360 (1978) · Zbl 0388.47022 [9] Kato, T.: Perturbation theory for linear operators. (1966) · Zbl 0148.12601 [10] Simon, B.: Schrödinger semigroup. Bull. amer. Math. soc. (N.S.) 7, 447-526 (1982) · Zbl 0524.35002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.