Advanced Search
Article Contents
Article Contents

Vector-valued Schrödinger operators in Lp-spaces

This work has been supported by the M.I.U.R. research project Prin 2015233N54 "Deterministic and Stochastic Evolution Equations". The third author is a member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM)

Abstract Full Text(HTML) Related Papers Cited by
  • In this paper we consider the vector-valued operator div$ (Q\nabla u)-Vu $ of Schrödinger type. Here $ V = (v_{ij}) $ is a nonnegative, locally bounded, matrix-valued function and $ Q $ is a symmetric, strictly elliptic matrix whose entries are bounded and continuously differentiable with bounded derivatives. Concerning the potential $ V $, we assume an that it is pointwise accretive and that its entries are in $ L^\infty_{{\rm loc}}( \mathbb{R}^d) $. Under these assumptions, we prove that a realization of the vector-valued Schrödinger operator generates a $ C_0 $-semigroup of contractions in $ L^p( \mathbb{R}^d; \mathbb{C}^m) $. Further properties are also investigated.

    Mathematics Subject Classification: Primary: 35K40, 47D08; Secondary: 47D06.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] D. AddonaL. AngiuliL. Lorenzi and G. Tessitore, On coupled systems of Kolmogorov equations with applications to stochastic differential games, ESAIM Control, Optim. Calc. of Var., 23 (2017), 937-976.  doi: 10.1051/cocv/2016019.
    [2] S. Agmon, The $L_{p}$ approach to the Dirichlet problem. Ⅰ. Regularity theorems, Ann. Scuola Norm. Sup. Pisa (3), 13 (1959), 405-448. 
    [3] L. AngiuliL. Lorenzi and D. Pallara, $L^p$-estimates for parabolic systems with unbounded coefficients coupled at zero and first order, J. Math. Anal. Appl., 444 (2016), 110-135.  doi: 10.1016/j.jmaa.2016.06.001.
    [4] V. BetzB. D. Goddard and S. Teufel, Superadiabatic transitions in quantum molecular dynamics, Proc. R. Soc. A, 465 (2009), 3553-3580.  doi: 10.1098/rspa.2009.0337.
    [5] G. M. Dall'Ara, Discreteness of the spectrum of Schrödinger operators with non-negative matrix-valued potentials, J. Funct. Anal., 268 (2015), 3649-3679.  doi: 10.1016/j.jfa.2014.10.007.
    [6] S. Delmonte and L. Lorenzi, On a class of weakly coupled systems of elliptic operators with unbounded coefficients, Milan J. Math., 79 (2011), 689-727.  doi: 10.1007/s00032-011-0170-7.
    [7] K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000.
    [8] S. Fornaro and L. Lorenzi, Generation results for elliptic operators with unbounded diffusion coefficients in $L^p$- and $C_b$-spaces, Discr. Cont. Dyn. Syst., 18 (2007), 747-772.  doi: 10.3934/dcds.2007.18.747.
    [9] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag Berlin, 2001.
    [10] T. Hansel and A. Rhandi, The Oseen-Navier-Stokes flow in the exterior of a rotating obstacle: The non-autonomous case, J. Rein. Angew. Math., 694 (2014), 1-26.  doi: 10.1515/crelle-2012-0113.
    [11] F. Haslinger and B. Helffer, Compactness of the solution operator to $\overline{\partial}$ in weighted $L^2$-spaces, J. Funct. Anal., 243 (2007), 679-697.  doi: 10.1016/j.jfa.2006.09.004.
    [12] M. HieberL. LorenziJ. Prüss and A. Rhandi, Global properties of generalized Ornstein-Uhlenbeck operators on $L^p(\mathbb{R}^N, \mathbb{R}^N)$ with more than linearly growing coefficients, J. Math. Anal. Appl., 350 (2009), 100-121.  doi: 10.1016/j.jmaa.2008.09.011.
    [13] M. HieberA. Rhandi and O. Sawada, The Navier-Stokes flow for globally Lipschitz continuous initial data, Kyoto Conference on the Navier-Stokes Equations and their Applications, Res. Inst. Math. Sci. (RIMS) Kkyroku Bessatsu, B1 (2007), 159-165. 
    [14] M. Hieber and O. Sawada, The Navier-Stokes equations in $\mathbb{R}^n$ with linearly growing initial data, Arch. Ration. Mech. Anal., 175 (2005), 269-285.  doi: 10.1007/s00205-004-0347-0.
    [15] T. Kato, On some Schrödinger operators with a singular complex potential, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 5 (1978), 105-114. 
    [16] M. Kunze, L. Lorenzi, A. Maichine and A. Rhandi, $L^p$-theory for Schrödinger systems, to appear in Math. Nachr, doi: 10.1002/mana.201800206, 2019. doi: 10.1002/mana.201800206.
    [17] L. Lorenzi, Analytical Methods for Kolmogorov Equations, Second edition, Monograph and research notes in Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2017.
    [18] L. Lorenzi and A. Rhandi, On Schrödinger type operators with unbounded coefficients: generation and heat kernel estimates, J. Evol. Equ., 15 (2015), 53-88.  doi: 10.1007/s00028-014-0249-z.
    [19] A. Maichine and A. Rhandi, On a polynomial scalar perturbation of a Schrödinger system in $L^p$-spaces, J. Math. Anal. Appl., 466 (2018), 655-675.  doi: 10.1016/j.jmaa.2018.06.014.
    [20] J. PrüssA. Rhandi and R. Schnaubelt, The domain of elliptic operators on $L^p(\mathbb{R}^d)$ with unbounded drift coefficients, Houston J. Math., 32 (2006), 563-576. 
    [21] K. Yosida, Functional Analysis, Springer-Verlag, Berlin Heidelberg, New York, 1980.
  • 加载中

Article Metrics

HTML views(576) PDF downloads(452) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint