## Illinois Journal of Mathematics

### Scattering length for stable processes

Bartłomiej Siudeja

#### Abstract

Let $0<\alpha<2$ and $X_t$ be the isotropic $\alpha$-stable Lévy process. We define scattering length $\Gamma(v)$ of a positive potential $v$. We use the scattering length to find estimates for the first eigenvalue of the Schrödinger operator of the “Neumann” fractional Laplacian in a cube with a potential $v$.

#### Article information

Source
Illinois J. Math., Volume 52, Number 2 (2008), 667-680.

Dates
First available in Project Euclid: 23 July 2009

https://projecteuclid.org/euclid.ijm/1248355357

Digital Object Identifier
doi:10.1215/ijm/1248355357

Mathematical Reviews number (MathSciNet)
MR2524659

Zentralblatt MATH identifier
1178.60037

Subjects
Primary: 60G52: Stable processes
Secondary: 31C15: Potentials and capacities

#### Citation

Siudeja, Bartłomiej. Scattering length for stable processes. Illinois J. Math. 52 (2008), no. 2, 667--680. doi:10.1215/ijm/1248355357. https://projecteuclid.org/euclid.ijm/1248355357

#### References

• B. Baumgartner, The existence of many-particle bound states despite a pair interaction with positive scattering length, J. Phys. A 30 (1997), L741–L747.
• R. M. Blumenthal and R. K. Getoor, Markov processes and potential theory, Academic Press, New York, 1968.
• K. Bogdan, K. Burdzy and Z.-Q. Chen, Censored stable processes, Probab. Theory Related Fields 127 (2003), 89–152.
• Z.-Q. Chen, Multidimensional symmetric stable processes, Korean J. Comput. Appl. Math. 6 (1999), 227–266.
• Z.-Q. Chen and T. Kumagai, Heat kernel estimates for stable-like processes on d-sets, Stochastic Process. Appl. 108 (2003), 27–62.
• E. B. Davies, Heat kernels and spectral theory, Cambridge Univ. Press, Cambridge, 1990.
• M. Kac, Probabilistic methods in some problems of scattering theory, Rocky Mountain J. of Math. 4 (1974), 511–537.
• M. Kac and J. Luttinger, Scattering length and capacity, Annales de l'Inst. Fourier 25 (1975), 317–321.
• K.-I. Sato, Lévy processes and infinitely divisible distributions, Translated from the 1990 Japanese original, Revised by the author, Cambridge Studies in Advanced Mathematics, vol. 68. Cambridge University Press, 1999.
• D. Stroock, The Kac approach to potential theory, I, J. Math. Mech. 16 (1967), 829–852.
• R. Szmytkowski, Analytical calculations of scattering lengths in atomic physics, J. Phys. A 28 (1995), 7333–7345.
• M. Taylor, Scattering length and perturbations of $-\Delta$ by positive potential, J. Math. Anal. Appl. 53 (1976), 291–312.
• M. Taylor, Scattering length of positive potential. Notes.