American Institute of Mathematical Sciences

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

Vector-valued Schrödinger operators in Lp-spaces

Markus Kunze 1, , Abdallah Maichine 2, and  Abdelaziz Rhandi 3,

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.

Keywords: System of PDE, Schrödinger operator, strongly continuous semigroup, domain characterization.
Mathematics Subject Classification: Primary: 35K40, 47D08; Secondary: 47D06.
Citation: Markus Kunze, Abdallah Maichine, Abdelaziz Rhandi. Vector-valued Schrödinger operators in Lp-spaces. Discrete & Continuous Dynamical Systems - S, 2020, 13 (5) : 1529-1541. doi: 10.3934/dcdss.2020086
References:
[1]

D. Addona, L. Angiuli, L. 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.  Google Scholar

[2]

S. Agmon, The $L_{p}$ approach to the Dirichlet problem. Ⅰ. Regularity theorems, Ann. Scuola Norm. Sup. Pisa (3), 13 (1959), 405-448.   Google Scholar

[3]

L. Angiuli, L. 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.  Google Scholar

[4]

V. Betz, B. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000.  Google Scholar

[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.  Google Scholar

[9]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag Berlin, 2001.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

M. Hieber, L. Lorenzi, J. 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.  Google Scholar

[13]

M. Hieber, A. 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.   Google Scholar

[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.  Google Scholar

[15]

T. Kato, On some Schrödinger operators with a singular complex potential, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 5 (1978), 105-114.   Google Scholar

[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.  Google Scholar

[17]

L. Lorenzi, Analytical Methods for Kolmogorov Equations, Second edition, Monograph and research notes in Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2017.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

J. Prüss, A. 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.   Google Scholar

[21]

K. Yosida, Functional Analysis, Springer-Verlag, Berlin Heidelberg, New York, 1980.  Google Scholar

show all references

References:
[1]

D. Addona, L. Angiuli, L. 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.  Google Scholar

[2]

S. Agmon, The $L_{p}$ approach to the Dirichlet problem. Ⅰ. Regularity theorems, Ann. Scuola Norm. Sup. Pisa (3), 13 (1959), 405-448.   Google Scholar

[3]

L. Angiuli, L. 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.  Google Scholar

[4]

V. Betz, B. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000.  Google Scholar

[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.  Google Scholar

[9]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag Berlin, 2001.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

M. Hieber, L. Lorenzi, J. 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.  Google Scholar

[13]

M. Hieber, A. 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.   Google Scholar

[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.  Google Scholar

[15]

T. Kato, On some Schrödinger operators with a singular complex potential, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 5 (1978), 105-114.   Google Scholar

[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.  Google Scholar

[17]

L. Lorenzi, Analytical Methods for Kolmogorov Equations, Second edition, Monograph and research notes in Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2017.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

J. Prüss, A. 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.   Google Scholar

[21]

K. Yosida, Functional Analysis, Springer-Verlag, Berlin Heidelberg, New York, 1980.  Google Scholar

[1]

Nguyen Dinh Cong, Roberta Fabbri. On the spectrum of the one-dimensional Schrödinger operator. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 541-554. doi: 10.3934/dcdsb.2008.9.541
[2]

Jianqing Chen. Sharp variational characterization and a Schrödinger equation with Hartree type nonlinearity. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1613-1628. doi: 10.3934/dcdss.2016066
[3]

Naoufel Ben Abdallah, Yongyong Cai, Francois Castella, Florian Méhats. Second order averaging for the nonlinear Schrödinger equation with strongly anisotropic potential. Kinetic & Related Models, 2011, 4 (4) : 831-856. doi: 10.3934/krm.2011.4.831
[4]

Maria Laura Delle Monache, Paola Goatin. A front tracking method for a strongly coupled PDE-ODE system with moving density constraints in traffic flow. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 435-447. doi: 10.3934/dcdss.2014.7.435
[5]

Hengguang Li, Jeffrey S. Ovall. A posteriori eigenvalue error estimation for a Schrödinger operator with inverse square potential. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1377-1391. doi: 10.3934/dcdsb.2015.20.1377
[6]

Xing-Bin Pan. An eigenvalue variation problem of magnetic Schrödinger operator in three dimensions. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 933-978. doi: 10.3934/dcds.2009.24.933
[7]

Ihyeok Seo. Carleman estimates for the Schrödinger operator and applications to unique continuation. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1013-1036. doi: 10.3934/cpaa.2012.11.1013
[8]

Valter Pohjola. An inverse problem for the magnetic Schrödinger operator on a half space with partial data. Inverse Problems & Imaging, 2014, 8 (4) : 1169-1189. doi: 10.3934/ipi.2014.8.1169
[9]

Ru-Yu Lai. Global uniqueness for an inverse problem for the magnetic Schrödinger operator. Inverse Problems & Imaging, 2011, 5 (1) : 59-73. doi: 10.3934/ipi.2011.5.59
[10]

Leyter Potenciano-Machado, Alberto Ruiz. Stability estimates for a magnetic Schrödinger operator with partial data. Inverse Problems & Imaging, 2018, 12 (6) : 1309-1342. doi: 10.3934/ipi.2018055
[11]

Joel Andersson, Leo Tzou. Stability for a magnetic Schrödinger operator on a Riemann surface with boundary. Inverse Problems & Imaging, 2018, 12 (1) : 1-28. doi: 10.3934/ipi.2018001
[12]

Chuang Zheng. Inverse problems for the fourth order Schrödinger equation on a finite domain. Mathematical Control & Related Fields, 2015, 5 (1) : 177-189. doi: 10.3934/mcrf.2015.5.177
[13]

George Avalos, Roberto Triggiani. Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 417-447. doi: 10.3934/dcdss.2009.2.417
[14]

Samir EL Mourchid. On a hypercylicity criterion for strongly continuous semigroups. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 271-275. doi: 10.3934/dcds.2005.13.271
[15]

Yohei Yamazaki. Transverse instability for a system of nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 565-588. doi: 10.3934/dcdsb.2014.19.565
[16]

Mei Ming. Weighted elliptic estimates for a mixed boundary system related to the Dirichlet-Neumann operator on a corner domain. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 6039-6067. doi: 10.3934/dcds.2019264
[17]

G. P. Trachanas, Nikolaos B. Zographopoulos. A strongly singular parabolic problem on an unbounded domain. Communications on Pure & Applied Analysis, 2014, 13 (2) : 789-809. doi: 10.3934/cpaa.2014.13.789
[18]

Sombuddha Bhattacharyya. An inverse problem for the magnetic Schrödinger operator on Riemannian manifolds from partial boundary data. Inverse Problems & Imaging, 2018, 12 (3) : 801-830. doi: 10.3934/ipi.2018034
[19]

Helge Krüger. Asymptotic of gaps at small coupling and applications of the skew-shift Schrödinger operator. Conference Publications, 2011, 2011 (Special) : 874-880. doi: 10.3934/proc.2011.2011.874
[20]

Angela A. Albanese, Xavier Barrachina, Elisabetta M. Mangino, Alfredo Peris. Distributional chaos for strongly continuous semigroups of operators. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2069-2082. doi: 10.3934/cpaa.2013.12.2069

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (151)
  • HTML views (405)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences