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

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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

Jussi Behrndt ^1, and A. F. M. ter Elst ^2,,

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.

Keywords: Dirichlet-to-Neumann graph, m-sectorial graph, form methods.

Mathematics Subject Classification: 35J57, 47F05.

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.
[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.

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