• Previous Article
    Intrinsic geometry and De Giorgi classes for certain anisotropic problems
  • DCDS-S Home
  • This Issue
  • Next Article
    Modeling and analysis of reactive multi-component two-phase flows with mass transfer and phase transition -- The isothermal incompressible case
2017, 10(4): 661-671. doi: 10.3934/dcdss.2017033

The Dirichlet-to-Neumann map for Schrödinger operators with complex potentials

1. 

Institut für Numerische Mathematik Technische Universität Graz Steyrergasse 30, A-8010 Graz, Austria

2. 

Department of Mathematics University of Auckland Private bag 92019, Auckland 1142, New Zealand

A.F.M ter Elst, E-mail address: terelst@math.auckland.ac.nz

Received  June 2016 Revised  December 2016 Published  April 2017

Let $\Omega \subset \mathbb{R}^d$ be a bounded open set with Lipschitz boundary and let $q \colon \Omega \to \mathbb{C}$ be a bounded complex potential. We study the Dirichlet-to-Neumann graph associated with the operator $- \Delta + q$ and we give an example in which it is not $m$-sectorial.
Citation: Jussi Behrndt, A. F. M. ter Elst. The Dirichlet-to-Neumann map for Schrödinger operators with complex potentials. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 661-671. doi: 10.3934/dcdss.2017033
References:
[1]

D. Alpay, J. Behrndt, Generalized Q-functions and Dirichlet-to-Neumann maps for elliptic differential operators, J. Funct. Anal., 257 (2009), 1666-1694.

[2]

W. Arendt, R. Chill, C. Seifert, H. Vogt and J. Voigt, Internet Seminar 18,2015.

[3]

W. Arendt, A. F. M. ter Elst, The Dirichlet-to-Neumann operator on rough domains, J. Diff. Eq., 251 (2011), 2100-2124.

[4]

Sectorial forms and degenerate differential operators, J. Operator Theory, 67 (2012), 33-72.

[5]

The Dirichlet-to-Neumann operator on exterior domains, Potential Anal. , 43 (2015), 313-340.

[6]

W. Arendt, A. F. M. ter Elst, J. B. Kennedy, M. Sauter, The Dirichlet-to-Neumann operator via hidden compactness, J. Funct. Anal., 266 (2014), 1757-1786.

[7]

W. Arendt, R. Mazzeo, Friedlander's eigenvalue inequalities and the Dirichlet-to-Neumann semigroup, Commun. Pure Appl. Anal., 11 (2012), 2201-2212.

[8]

J. Behrndt, A. F. M. ter Elst, Dirichlet-to-Neumann maps on bounded Lipschitz domains, J. Diff. Eq., 259 (2015), 5903-5926.

[9]

J. Behrndt, F. Gesztesy, H. Holden, R. Nichols, Dirichlet-to-Neumann maps, abstract Weyl-Titchmarsh M-functions, and a generalized index of unbounded meromorphic operatorvalued functions, J. Diff. Eq., 261 (2016), 3551-3587.

[10]

J. Behrndt, M. Langer, Boundary value problems for elliptic partial differential operators on bounded domains, J. Funct. Anal., 243 (2007), 536-565.

[11]

Elliptic operators, Dirichlet-to-Neumann maps and quasi boundary triples, in Operator Methods for Boundary Value Problems, London Math. Soc. Lecture Note Ser. , 404, Cambridge Univ. Press, Cambridge, 2012,121-160.

[12]

J. Behrndt, J. Rohleder, Spectral analysis of selfadjoint elliptic differential operators, Dirichlet-to-Neumann maps, and abstract Weyl functions, Adv. Math., 285 (2015), 1301-1338.

[13]

B. M. Brown, G. Grubb, I. G. Wood, M-functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems, Math. Nachr., 282 (2009), 314-347.

[14]

A. F. M. ter Elst, E. -M. Ouhabaz, Analysis of the heat kernel of the Dirichlet-to-Neumann operator, J. Funct. Anal., 267 (2014), 4066-4109.

[15]

Convergence of the Dirichlet-to-Neumann operator on varying domains, in Operator Semigroups Meet Complex Analysis, Harmonic Analysis and Mathematical Physics, Operator Theory: Advances and Applications, 250, Birkhäuser, 2015,147-154.

[16]

F. Gesztesy and M. Mitrea, Generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains, in Perspectives in Partial Differential Equations, Harmonic Analysis and Applications, Proc. Sympos. Pure Math. , 79, Amer. Math. Soc. , Providence, RI, 2008,105-173.

[17]

A description of all self-adjoint extensions of the Laplacian and Krein-type resolvent formulas on non-smooth domains, J. Anal. Math. , 113 (2011), 53-172.

[18]

F. Gesztesy, M. Mitrea, M. Zinchenko, Variations on a theme of Jost and Pais, J. Funct. Anal., 253 (2007), 399-448.

[19]

On Dirichlet-to-Neumann maps and some applications to modified Fredholm determinants, in Methods of Spectral Analysis in Mathematical Physics, Oper. Theory Adv. Appl. , 186, Birkhäuser Verlag, Basel, 2009,191-215.

[20]

H. Gimperlein, G. Grubb, Heat kernel estimates for pseudodifferential operators, fractional Laplacians and Dirichlet-to-Neumann operators, J. Evol. Equ., 14 (2014), 49-83.

[21]

D. Jerison, C. E. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal., 130 (1995), 161-219.

[22]

T. Kato, Perturbation Theory for Linear Operators, Second edition, Grundlehren der mathematischen Wissenschaften, 132, Springer-Verlag, Berlin etc. , 1980.

[23]

M. M. Malamud, Spectral theory of elliptic operators in exterior domains, Russ. J. Math. Phys., 17 (2010), 96-125.

[24]

A. B. Mikhailova, B. S. Pavlov, L. V. Prokhorov, Intermediate Hamiltonian via Glazman's splitting and analytic perturbation for meromorphic matrix-functions, Math. Nachr., 280 (2007), 1376-1416.

[25]

A. B. Mikhailova, B. S. Pavlov and V. I. Ryzhii, Dirichlet-to-Neumann techniques for the plasma-waves in a slot-diode, in Operator Theory, Analysis and Mathematical Physics, Oper. Theory Adv. Appl. , 174, Birkhäuser, Basel, 2007, 74-103.

[26]

O. Post, Boundary pairs associated with quadratic forms, Math. Nachr., 289 (2016), 1052-1099.

[27]

M. Warma, A fractional Dirichlet-to-Neumann operator on bounded Lipschitz domains, Commun. Pure Appl. Anal., 14 (2015), 2043-2067.

show all references

References:
[1]

D. Alpay, J. Behrndt, Generalized Q-functions and Dirichlet-to-Neumann maps for elliptic differential operators, J. Funct. Anal., 257 (2009), 1666-1694.

[2]

W. Arendt, R. Chill, C. Seifert, H. Vogt and J. Voigt, Internet Seminar 18,2015.

[3]

W. Arendt, A. F. M. ter Elst, The Dirichlet-to-Neumann operator on rough domains, J. Diff. Eq., 251 (2011), 2100-2124.

[4]

Sectorial forms and degenerate differential operators, J. Operator Theory, 67 (2012), 33-72.

[5]

The Dirichlet-to-Neumann operator on exterior domains, Potential Anal. , 43 (2015), 313-340.

[6]

W. Arendt, A. F. M. ter Elst, J. B. Kennedy, M. Sauter, The Dirichlet-to-Neumann operator via hidden compactness, J. Funct. Anal., 266 (2014), 1757-1786.

[7]

W. Arendt, R. Mazzeo, Friedlander's eigenvalue inequalities and the Dirichlet-to-Neumann semigroup, Commun. Pure Appl. Anal., 11 (2012), 2201-2212.

[8]

J. Behrndt, A. F. M. ter Elst, Dirichlet-to-Neumann maps on bounded Lipschitz domains, J. Diff. Eq., 259 (2015), 5903-5926.

[9]

J. Behrndt, F. Gesztesy, H. Holden, R. Nichols, Dirichlet-to-Neumann maps, abstract Weyl-Titchmarsh M-functions, and a generalized index of unbounded meromorphic operatorvalued functions, J. Diff. Eq., 261 (2016), 3551-3587.

[10]

J. Behrndt, M. Langer, Boundary value problems for elliptic partial differential operators on bounded domains, J. Funct. Anal., 243 (2007), 536-565.

[11]

Elliptic operators, Dirichlet-to-Neumann maps and quasi boundary triples, in Operator Methods for Boundary Value Problems, London Math. Soc. Lecture Note Ser. , 404, Cambridge Univ. Press, Cambridge, 2012,121-160.

[12]

J. Behrndt, J. Rohleder, Spectral analysis of selfadjoint elliptic differential operators, Dirichlet-to-Neumann maps, and abstract Weyl functions, Adv. Math., 285 (2015), 1301-1338.

[13]

B. M. Brown, G. Grubb, I. G. Wood, M-functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems, Math. Nachr., 282 (2009), 314-347.

[14]

A. F. M. ter Elst, E. -M. Ouhabaz, Analysis of the heat kernel of the Dirichlet-to-Neumann operator, J. Funct. Anal., 267 (2014), 4066-4109.

[15]

Convergence of the Dirichlet-to-Neumann operator on varying domains, in Operator Semigroups Meet Complex Analysis, Harmonic Analysis and Mathematical Physics, Operator Theory: Advances and Applications, 250, Birkhäuser, 2015,147-154.

[16]

F. Gesztesy and M. Mitrea, Generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains, in Perspectives in Partial Differential Equations, Harmonic Analysis and Applications, Proc. Sympos. Pure Math. , 79, Amer. Math. Soc. , Providence, RI, 2008,105-173.

[17]

A description of all self-adjoint extensions of the Laplacian and Krein-type resolvent formulas on non-smooth domains, J. Anal. Math. , 113 (2011), 53-172.

[18]

F. Gesztesy, M. Mitrea, M. Zinchenko, Variations on a theme of Jost and Pais, J. Funct. Anal., 253 (2007), 399-448.

[19]

On Dirichlet-to-Neumann maps and some applications to modified Fredholm determinants, in Methods of Spectral Analysis in Mathematical Physics, Oper. Theory Adv. Appl. , 186, Birkhäuser Verlag, Basel, 2009,191-215.

[20]

H. Gimperlein, G. Grubb, Heat kernel estimates for pseudodifferential operators, fractional Laplacians and Dirichlet-to-Neumann operators, J. Evol. Equ., 14 (2014), 49-83.

[21]

D. Jerison, C. E. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal., 130 (1995), 161-219.

[22]

T. Kato, Perturbation Theory for Linear Operators, Second edition, Grundlehren der mathematischen Wissenschaften, 132, Springer-Verlag, Berlin etc. , 1980.

[23]

M. M. Malamud, Spectral theory of elliptic operators in exterior domains, Russ. J. Math. Phys., 17 (2010), 96-125.

[24]

A. B. Mikhailova, B. S. Pavlov, L. V. Prokhorov, Intermediate Hamiltonian via Glazman's splitting and analytic perturbation for meromorphic matrix-functions, Math. Nachr., 280 (2007), 1376-1416.

[25]

A. B. Mikhailova, B. S. Pavlov and V. I. Ryzhii, Dirichlet-to-Neumann techniques for the plasma-waves in a slot-diode, in Operator Theory, Analysis and Mathematical Physics, Oper. Theory Adv. Appl. , 174, Birkhäuser, Basel, 2007, 74-103.

[26]

O. Post, Boundary pairs associated with quadratic forms, Math. Nachr., 289 (2016), 1052-1099.

[27]

M. Warma, A fractional Dirichlet-to-Neumann operator on bounded Lipschitz domains, Commun. Pure Appl. Anal., 14 (2015), 2043-2067.

[1]

Mahamadi Warma. A fractional Dirichlet-to-Neumann operator on bounded Lipschitz domains. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2043-2067. doi: 10.3934/cpaa.2015.14.2043

[2]

Mourad Bellassoued, David Dos Santos Ferreira. Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map. Inverse Problems & Imaging, 2011, 5 (4) : 745-773. doi: 10.3934/ipi.2011.5.745

[3]

Wolfgang Arendt, Rafe Mazzeo. Friedlander's eigenvalue inequalities and the Dirichlet-to-Neumann semigroup. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2201-2212. doi: 10.3934/cpaa.2012.11.2201

[4]

Victor Isakov, Jenn-Nan Wang. Increasing stability for determining the potential in the Schrödinger equation with attenuation from the Dirichlet-to-Neumann map. Inverse Problems & Imaging, 2014, 8 (4) : 1139-1150. doi: 10.3934/ipi.2014.8.1139

[5]

Oded Schramm. Hyperfinite graph limits. Electronic Research Announcements, 2008, 15: 17-23. doi: 10.3934/era.2008.15.17

[6]

J. William Hoffman. Remarks on the zeta function of a graph. Conference Publications, 2003, 2003 (Special) : 413-422. doi: 10.3934/proc.2003.2003.413

[7]

John Kieffer and En-hui Yang. Ergodic behavior of graph entropy. Electronic Research Announcements, 1997, 3: 11-16.

[8]

Eric Babson and Dmitry N. Kozlov. Topological obstructions to graph colorings. Electronic Research Announcements, 2003, 9: 61-68.

[9]

Daniel Roggen, Martin Wirz, Gerhard Tröster, Dirk Helbing. Recognition of crowd behavior from mobile sensors with pattern analysis and graph clustering methods. Networks & Heterogeneous Media, 2011, 6 (3) : 521-544. doi: 10.3934/nhm.2011.6.521

[10]

Mario Roy, Mariusz Urbański. Random graph directed Markov systems. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 261-298. doi: 10.3934/dcds.2011.30.261

[11]

Dominique Zosso, Braxton Osting. A minimal surface criterion for graph partitioning. Inverse Problems & Imaging, 2016, 10 (4) : 1149-1180. doi: 10.3934/ipi.2016036

[12]

Mario Roy, Mariusz Urbański. Multifractal analysis for conformal graph directed Markov systems. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 627-650. doi: 10.3934/dcds.2009.25.627

[13]

Mirela Domijan, Markus Kirkilionis. Graph theory and qualitative analysis of reaction networks. Networks & Heterogeneous Media, 2008, 3 (2) : 295-322. doi: 10.3934/nhm.2008.3.295

[14]

Matthew Macauley, Henning S. Mortveit. Update sequence stability in graph dynamical systems. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1533-1541. doi: 10.3934/dcdss.2011.4.1533

[15]

Deena Schmidt, Janet Best, Mark S. Blumberg. Random graph and stochastic process contributions to network dynamics. Conference Publications, 2011, 2011 (Special) : 1279-1288. doi: 10.3934/proc.2011.2011.1279

[16]

Lasse Kliemann, Elmira Shirazi Sheykhdarabadi, Anand Srivastav. Price of anarchy for graph coloring games with concave payoff. Journal of Dynamics & Games, 2017, 4 (1) : 41-58. doi: 10.3934/jdg.2017003

[17]

M. D. König, Stefano Battiston, M. Napoletano, F. Schweitzer. On algebraic graph theory and the dynamics of innovation networks. Networks & Heterogeneous Media, 2008, 3 (2) : 201-219. doi: 10.3934/nhm.2008.3.201

[18]

Chun-Xiang Guo, Guo Qiang, Jin Mao-Zhu, Zhihan Lv. Dynamic systems based on preference graph and distance. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1139-1154. doi: 10.3934/dcdss.2015.8.1139

[19]

Sergei Avdonin, Jonathan Bell. Determining a distributed conductance parameter for a neuronal cable model defined on a tree graph. Inverse Problems & Imaging, 2015, 9 (3) : 645-659. doi: 10.3934/ipi.2015.9.645

[20]

Ana Jofre, Lan-Xi Dong, Ha Phuong Vu, Steve Szigeti, Sara Diamond. Rendering website traffic data into interactive taste graph visualizations. Big Data & Information Analytics, 2017, 2 (2) : 107-118. doi: 10.3934/bdia.2017003

2016 Impact Factor: 0.781

Metrics

  • PDF downloads (0)
  • HTML views (1)
  • Cited by (0)

Other articles
by authors

[Back to Top]