## Acta Mathematica

### Quantum diffusion of the random Schrödinger evolution in the scaling limit

#### Abstract

We consider random Schrödinger equations on Rd for d ≽ 3 with a homogeneous Anderson–Poisson type random potential. Denote by λ the coupling constant and $\psi_t$ the solution with initial data $\psi_0$. The space and time variables scale as $x\sim\lambda ^{{ - 2 - \varkappa/2}} {\text{ and }}t\sim\lambda ^{{ - 2 - \varkappa}} {\text{ with }}0 < \varkappa < \varkappa_{0} {\left( d \right)}$. We prove that, in the limit λ → 0, the expectation of the Wigner distribution of $\psi_t$ converges weakly to the solution of a heat equation in the space variable x for arbitrary L2 initial data.

The proof is based on analyzing the phase cancellations of multiple scatterings on the random potential by expanding the propagator into a sum of Feynman graphs. In this paper we consider the non-recollision graphs and prove that the amplitude of the non-ladder diagrams is smaller than their “naive size” by an extra λc factor per non-(anti)ladder vertex for some c > 0. This is the first rigorous result showing that the improvement over the naive estimates on the Feynman graphs grows as a power of the small parameter with the exponent depending linearly on the number of vertices. This estimate allows us to prove the convergence of the perturbation series.

#### Note

The first auhor was partially supported by NSF grant DMS-0307295 and MacArthur Fellowship. The third author was partially supported by NSF grant DMS-0200235 and EU-IHP Network “Analysis and Quantum” HPRN-CT-2002-0027.

#### Article information

Source
Acta Math., Volume 200, Number 2 (2008), 211-277.

Dates
Revised: 2 April 2007
First available in Project Euclid: 31 January 2017

https://projecteuclid.org/euclid.acta/1485891980

Digital Object Identifier
doi:10.1007/s11511-008-0027-2

Mathematical Reviews number (MathSciNet)
MR2413135

Zentralblatt MATH identifier
1155.82015

Rights

#### Citation

Erdős, László; Salmhofer, Manfred; Yau, Horng-Tzer. Quantum diffusion of the random Schrödinger evolution in the scaling limit. Acta Math. 200 (2008), no. 2, 211--277. doi:10.1007/s11511-008-0027-2. https://projecteuclid.org/euclid.acta/1485891980

#### References

• A izenman, M. & M olchanov, S., Localization at large disorder and at extreme energies: an elementary derivation. Comm. Math. Phys., 157 (1993), 245–278.
• A izenman, M., S ims, R. & W arzel, S., Absolutely continuous spectra of quantum tree graphs with weak disorder. Comm. Math. Phys., 264 (2006), 371–389.
• A nderson, P., Absences of diffusion in certain random lattices. Phys. Rev., 109 (1958), 1492–1505.
• B oldrighini, C., B unimovich, L. A. & S inaĭ, Y. G., On the Boltzmann equation for the Lorentz gas. J. Stat. Phys., 32 (1983), 477–501.
• B ourgain, J., Random lattice Schrödinger operators with decaying potential: some higher dimensional phenomena, in Geometric Aspects of Functional Analysis, Lecture Notes in Math., 1807, pp. 70–98. Springer, Berlin–Heidelberg, 2003.
• B rown, R., A brief account of microscopical observations made in the months of June, July, and August 1827, on the particles contained in the pollen of plants and on the general existence of active molecules in organic and inorganic bodies. Philosophical Magazine (2), 4 (1828), 161–173.
• — Additional remarks on active molecules. Philosophical Magazine (2), 6 (1829), 161–166.
• B rydges, D., D imock, J. & H urd, T. R., The short distance behavior of (ϕ4)3. Comm. Math. Phys., 172 (1995), 143–186.
• B rydges, D. & Y au, H.-T., Grad ϕ perturbations of massless Gaussian fields. Comm. Math. Phys., 129 (1990), 351–392.
• B unimovich, L. A. & S inaĭ, Y.G., Statistical properties of Lorentz gas with periodic configuration of scatterers. Comm. Math. Phys., 78 (1980/81), 479–497.
• C hen, T., Localization lengths and Boltzmann limit for the Anderson model at small disorders in dimension 3. J. Stat. Phys., 120 (2005), 279–337.
• D enisov, S.A., Absolutely continuous spectrum of multidimensional Schrödinger operator. Int. Math. Res. Not., 74 (2004), 3963–3982.
• D ürr, D., G oldstein, S. & L ebowitz, J. L., A mechanical model of Brownian motion. Comm. Math. Phys., 78 (1980/81), 507–530.
• — Asymptotic motion of a classical particle in a random potential in two dimensions: Landau model. Comm. Math. Phys., 113 (1987), 209–230.
• E instein, A., Zur Theorie der Brownschen Bewegung. Ann. der Physik, 19 (1906), 371–381.
• E rdős, L., Linear Boltzmann equation as the long time dynamics of an electron weakly coupled to a phonon field. J. Stat. Phys., 107 (2002), 1043–1127.
• E rdős, L., S almhofer, M. & Y au, H.-T., Towards the quantum Brownian motion, in Mathematical Physics of Quantum Mechanics, Lecture Notes in Physics, 690, pp. 233–257. Springer, Berlin–Heidelberg, 2006.
• — Quantum diffusion for the Anderson model in the scaling limit. Ann. Henri Poincaré, 8 (2007), 621–685.
• — Quantum diffusion of the random Schrödinger evolution in the scaling limit. II. The recollision diagrams. Comm. Math. Phys., 271 (2007), 1–53.
• E rdős, L. & Y au, H.-T., Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation. Comm. Pure Appl. Math., 53 (2000), 667–735.
• F eldman, J., K nörrer, H. & T rubowitz, E., A representation for fermionic correlation functions. Comm. Math. Phys., 195 (1998), 465–493.
• — Convergence of perturbation expansions in Fermionic models. II. Overlapping loops. Comm. Math. Phys., 247 (2004), 243–319.
• F eldman, J., M agnen, J., R ivasseau, V. & S énéor, R., Bounds on completely convergent Euclidean Feynman graphs. Comm. Math. Phys., 98 (1985), 273–288.
• — Bounds on renormalized Feynman graphs. Comm. Math. Phys., 100 (1985), 23–55.
• — A renormalizable field theory: the massive Gross–Neveu model in two dimensions. Comm. Math. Phys., 103 (1986), 67–103.
• — Construction and Borel summability of infrared Φ44 by a phase space expansion. Comm. Math. Phys., 109 (1987), 437–480.
• F eldman, J., S almhofer, M. & T rubowitz, E., Perturbation theory around nonnested Fermi surfaces. I. Keeping the Fermi surface fixed. J. Stat. Phys., 84 (1996), 1209–1336.
• — Regularity of interacting nonspherical Fermi surfaces: the full self-energy. Comm. Pure Appl. Math., 52 (1999), 273–324.
• F roese, R., H asler, D. & S pitzer, W., Transfer matrices, hyperbolic geometry and absolutely continuous spectrum for some discrete Schrödinger operators on graphs. J. Funct. Anal., 230 (2006), 184–221.
• F röhlich, J. & S pencer, T., Absence of diffusion in the Anderson tight binding model for large disorder or low energy. Comm. Math. Phys., 88 (1983), 151–184.
• G allavotti, G., Rigorous theory of the Boltzmann equation in the Lorentz model. Nota interna n. 358, Physics Department, Università “La Sapienza”, Roma, (1972), pp. 1–9. mp arc@math.utexas.edu, #93–304.
• G awędzki, K. & K upiainen, A., Gross–Neveu model through convergent perturbation expansions. Comm. Math. Phys., 102 (1985), 1–30.
• — Massless lattice φ44 theory: rigorous control of a renormalizable asymptotically free model. Comm. Math. Phys., 99 (1985), 197–252.
• G ol′dsheĭd, I. J., M olchanov, S.A. & P astur, L.A., A random homogeneous Schrödinger operator has a pure point spectrum. Funktsional. Anal. i Prilozhen., 11 (1977), 1–10 (Russian); English translation in Funct. Anal. Appl., 11 (1977), 1–8.
• H o, T.G., L andau, L. J. & W ilkins, A. J., On the weak coupling limit for a Fermi gas in a random potential. Rev. Math. Phys., 5 (1993), 209–298.
• K esten, H. & P apanicolaou, G. C., A limit theorem for stochastic acceleration. Comm. Math. Phys., 78 (1980), 19–63.
• K irsch, W. & M artinelli, F., On the essential selfadjointness of stochastic Schrödinger operators. Duke Math. J., 50 (1983), 1255–1260.
• K lein, A., Absolutely continuous spectrum in the Anderson model on the Bethe lattice. Math. Res. Lett., 1 (1994), 399–407.
• K omorowski, T. & R yzhik, L., Diffusion in a weakly random Hamiltonian flow. Comm. Math. Phys., 263 (2006), 277–323.
• L anford, O. E., On a derivation of the Boltzmann equation, in International Conference on Dynamical Systems in Mathematical Physics (Rennes, 1975), Astérisque, 40, pp. 117–137. Soc. Math. France, Paris, 1976.
• L ee, P. A. & R amakrishnan, T. V., Disordered electronic systems. Rev. Mod. Phys., 57 (1985), 287–337.
• L ukkarinen, J. & S pohn, H., Kinetic limit for wave propagation in a random medium. Arch. Ration. Mech. Anal., 183 (2007), 93–162.
• P errin, J.B., Mouvement brownien et réalité moléculaire. Annales de chimie et de physiqe, VIII 18 (1909), 5–114.
• R odnianski, I. & S chlag, W., Classical and quantum scattering for a class of long range random potentials. Int. Math. Res. Not., 5 (2003), 243–300.
• S almhofer, M. & W ieczerkowski, C., Positivity and convergence in fermionic quantum field theory. J. Stat. Phys., 99 (2000), 557–586.
• S chlag, W., S hubin, C. & W olff, T., Frequency concentration and location lengths for the Anderson model at small disorders. J. Anal. Math., 88 (2002), 173–220.
• S eiler, E., Gauge Theories as a Problem of Constructive Quantum Field Theory and Statistical Mechanics. Lecture Notes in Physics, 159. Springer, Berlin–Heidelberg, 1982.
• S pohn, H., Derivation of the transport equation for electrons moving through random impurities. J. Stat. Phys., 17 (1977), 385–412.
• — The Lorentz process converges to a random flight process. Comm. Math. Phys., 60 (1978), 277–290.
• V ollhardt, D. & W ölfle, P., Diagrammatic, self-consistent treatment of the Anderson localization problem in d≤2 dimensions. Phys. Rev. B, 22 (1980), 4666–4679.