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August  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

* Corresponding author: A.F.M ter Elst

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

W. ArendtA. F. M. ter ElstJ. B. Kennedy and M. Sauter, The Dirichlet-to-Neumann operator via hidden compactness, J. Funct. Anal., 266 (2014), 1757-1786.   Google Scholar

[7]

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

[8]

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

[9]

J. BehrndtF. GesztesyH. 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.   Google Scholar

[10]

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

[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. Google Scholar

[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.   Google Scholar

[13]

B. M. BrownG. 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.   Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[18]

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

[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. Google Scholar

[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.   Google Scholar

[21]

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

[22]

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

[23]

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

[24]

A. B. MikhailovaB. 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.   Google Scholar

[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. Google Scholar

[26]

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

[27]

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

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.   Google Scholar

[2]

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

[3]

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

[4]

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

[5]

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

[6]

W. ArendtA. F. M. ter ElstJ. B. Kennedy and M. Sauter, The Dirichlet-to-Neumann operator via hidden compactness, J. Funct. Anal., 266 (2014), 1757-1786.   Google Scholar

[7]

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

[8]

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

[9]

J. BehrndtF. GesztesyH. 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.   Google Scholar

[10]

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

[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. Google Scholar

[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.   Google Scholar

[13]

B. M. BrownG. 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.   Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[18]

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

[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. Google Scholar

[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.   Google Scholar

[21]

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

[22]

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

[23]

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

[24]

A. B. MikhailovaB. 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.   Google Scholar

[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. Google Scholar

[26]

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

[27]

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

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