-
Previous Article
Modeling and analysis of reactive multi-component two-phase flows with mass transfer and phase transition the isothermal incompressible case
- DCDS-S Home
- This Issue
-
Next Article
Intrinsic geometry and De Giorgi classes for certain anisotropic problems
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 |
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.
References:
| [1] |
D. Alpay and 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 and 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 and M. Sauter,
The Dirichlet-to-Neumann operator via hidden compactness, J. Funct. Anal., 266 (2014), 1757-1786.
|
| [7] |
W. Arendt and R. Mazzeo,
Friedlander's eigenvalue inequalities and the Dirichlet-to-Neumann semigroup, Commun. Pure Appl. Anal., 11 (2012), 2201-2212.
|
| [8] |
J. Behrndt and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and M. Sauter,
The Dirichlet-to-Neumann operator via hidden compactness, J. Funct. Anal., 266 (2014), 1757-1786.
|
| [7] |
W. Arendt and R. Mazzeo,
Friedlander's eigenvalue inequalities and the Dirichlet-to-Neumann semigroup, Commun. Pure Appl. Anal., 11 (2012), 2201-2212.
|
| [8] |
J. Behrndt and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and Applied Analysis, 2012, 11 (6) : 2201-2212. doi: 10.3934/cpaa.2012.11.2201 |
| [4] |
Kevin Arfi, Anna Rozanova-Pierrat. Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by d-sets. Discrete and Continuous Dynamical Systems - S, 2019, 12 (1) : 1-26. doi: 10.3934/dcdss.2019001 |
| [5] |
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 and Imaging, 2014, 8 (4) : 1139-1150. doi: 10.3934/ipi.2014.8.1139 |
| [6] |
Oleg Yu. Imanuvilov, Masahiro Yamamoto. Stability for determination of Riemannian metrics by spectral data and Dirichlet-to-Neumann map limited on arbitrary subboundary. Inverse Problems and Imaging, 2019, 13 (6) : 1213-1258. doi: 10.3934/ipi.2019054 |
| [7] |
Mourad Bellassoued, Zouhour Rezig. Recovery of transversal metric tensor in the Schrödinger equation from the Dirichlet-to-Neumann map. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1061-1084. doi: 10.3934/dcdss.2021158 |
| [8] |
Eric Babson and Dmitry N. Kozlov. Topological obstructions to graph colorings. Electronic Research Announcements, 2003, 9: 61-68. |
| [9] |
Oded Schramm. Hyperfinite graph limits. Electronic Research Announcements, 2008, 15: 17-23. doi: 10.3934/era.2008.15.17 |
| [10] |
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 |
| [11] |
John Kieffer and En-hui Yang. Ergodic behavior of graph entropy. Electronic Research Announcements, 1997, 3: 11-16. |
| [12] |
Roberto De Leo, James A. Yorke. The graph of the logistic map is a tower. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5243-5269. doi: 10.3934/dcds.2021075 |
| [13] |
Daniel Roggen, Martin Wirz, Gerhard Tröster, Dirk Helbing. Recognition of crowd behavior from mobile sensors with pattern analysis and graph clustering methods. Networks and Heterogeneous Media, 2011, 6 (3) : 521-544. doi: 10.3934/nhm.2011.6.521 |
| [14] |
Roy H. Goodman. NLS bifurcations on the bowtie combinatorial graph and the dumbbell metric graph. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 2203-2232. doi: 10.3934/dcds.2019093 |
| [15] |
Mario Roy, Mariusz Urbański. Random graph directed Markov systems. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 261-298. doi: 10.3934/dcds.2011.30.261 |
| [16] |
Rui Wang, Rundong Zhao, Emily Ribando-Gros, Jiahui Chen, Yiying Tong, Guo-Wei Wei. HERMES: Persistent spectral graph software. Foundations of Data Science, 2021, 3 (1) : 67-97. doi: 10.3934/fods.2021006 |
| [17] |
Dominique Zosso, Braxton Osting. A minimal surface criterion for graph partitioning. Inverse Problems and Imaging, 2016, 10 (4) : 1149-1180. doi: 10.3934/ipi.2016036 |
| [18] |
Mario Jorge Dias Carneiro, Rafael O. Ruggiero. On the graph theorem for Lagrangian minimizing tori. Discrete and Continuous Dynamical Systems, 2018, 38 (12) : 6029-6045. doi: 10.3934/dcds.2018260 |
| [19] |
Gökhan Mutlu. On the quotient quantum graph with respect to the regular representation. Communications on Pure and Applied Analysis, 2021, 20 (2) : 885-902. doi: 10.3934/cpaa.2020295 |
| [20] |
Chun-Xiang Guo, Guo Qiang, Jin Mao-Zhu, Zhihan Lv. Dynamic systems based on preference graph and distance. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1139-1154. doi: 10.3934/dcdss.2015.8.1139 |
2020 Impact Factor: 2.425
Tools
Metrics
Other articles
by authors
[Back to Top]