May  2020, 13(5): 1529-1541. doi: 10.3934/dcdss.2020086

Vector-valued Schrödinger operators in Lp-spaces

1. 

Fachbereich Mathematik und Statistik, Universität Konstanz, 78457 Konstanz, Germany

2. 

Dipartimento di Matematica, Università degli Studi di Salerno, via Giovanni Paolo Ⅱ, 132, 84084, Fisciano (Sa), Italy

3. 

Dipartimento di Ingegneria dell'Informazione, Ingegneria Elettrica e Matematica Applicata, Università degli Studi di Salerno, via Giovanni Paolo Ⅱ, 132, 84084, Fisciano (Sa), Italy

Received  February 2018 Revised  November 2018 Published  June 2019

Fund Project: 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)

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.

Citation: Markus Kunze, Abdallah Maichine, Abdelaziz Rhandi. Vector-valued Schrödinger operators in Lp-spaces. Discrete and Continuous Dynamical Systems - S, 2020, 13 (5) : 1529-1541. doi: 10.3934/dcdss.2020086
References:
[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.

show all references

References:
[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.

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