# Boundary Data Maps for Schrödinger Operators on a Compact Interval

S. Clark; F. Gesztesy; M. Mitrea

Mathematical Modelling of Natural Phenomena (2010)

- Volume: 5, Issue: 4, page 73-121
- ISSN: 0973-5348

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topClark, S., Gesztesy, F., and Mitrea, M.. "Boundary Data Maps for Schrödinger Operators on a Compact Interval." Mathematical Modelling of Natural Phenomena 5.4 (2010): 73-121. <http://eudml.org/doc/197614>.

@article{Clark2010,

abstract = {We provide a systematic study of boundary data maps, that is, 2 × 2 matrix-valued
Dirichlet-to-Neumann and more generally, Robin-to-Robin maps, associated with
one-dimensional Schrödinger operators on a compact interval [0, R] with
separated boundary conditions at 0 and R. Most of our results are
formulated in the non-self-adjoint context. Our principal results include explicit representations of these boundary data maps in
terms of the resolvent of the underlying Schrödinger operator and the associated boundary
trace maps, Krein-type resolvent formulas relating Schrödinger operators corresponding to
different (separated) boundary conditions, and a derivation of the Herglotz property of
boundary data maps (up to right multiplication by an appropriate diagonal matrix) in the
special self-adjoint case.},

author = {Clark, S., Gesztesy, F., Mitrea, M.},

journal = {Mathematical Modelling of Natural Phenomena},

keywords = {(non-self-adjoint) Schrödinger operators on a compact interval; separated boundary conditions; boundary data maps; Robin-to-Robin maps; linear fractional transformations; Krein-type resolvent formulas; Kreĭn-type resolvent formulas},

language = {eng},

month = {5},

number = {4},

pages = {73-121},

publisher = {EDP Sciences},

title = {Boundary Data Maps for Schrödinger Operators on a Compact Interval},

url = {http://eudml.org/doc/197614},

volume = {5},

year = {2010},

}

TY - JOUR

AU - Clark, S.

AU - Gesztesy, F.

AU - Mitrea, M.

TI - Boundary Data Maps for Schrödinger Operators on a Compact Interval

JO - Mathematical Modelling of Natural Phenomena

DA - 2010/5//

PB - EDP Sciences

VL - 5

IS - 4

SP - 73

EP - 121

AB - We provide a systematic study of boundary data maps, that is, 2 × 2 matrix-valued
Dirichlet-to-Neumann and more generally, Robin-to-Robin maps, associated with
one-dimensional Schrödinger operators on a compact interval [0, R] with
separated boundary conditions at 0 and R. Most of our results are
formulated in the non-self-adjoint context. Our principal results include explicit representations of these boundary data maps in
terms of the resolvent of the underlying Schrödinger operator and the associated boundary
trace maps, Krein-type resolvent formulas relating Schrödinger operators corresponding to
different (separated) boundary conditions, and a derivation of the Herglotz property of
boundary data maps (up to right multiplication by an appropriate diagonal matrix) in the
special self-adjoint case.

LA - eng

KW - (non-self-adjoint) Schrödinger operators on a compact interval; separated boundary conditions; boundary data maps; Robin-to-Robin maps; linear fractional transformations; Krein-type resolvent formulas; Kreĭn-type resolvent formulas

UR - http://eudml.org/doc/197614

ER -

## References

top- N. I. Akhiezer, I. M. Glazman. Theory of linear operators in Hilbert space, Volume II. Pitman, Boston, 1981. Zbl0467.47001
- S. Albeverio, J. F. Brasche, M. M. Malamud, H. Neidhardt. Inverse spectral theory for symmetric operators with several gaps: scalar-type Weyl functions. J. Funct. Anal., 228 (2005), 144–188. Zbl1083.47020
- D. Alpay, J. Behrndt. GeneralizedQ-functions and Dirichlet-to-Neumann maps for elliptic differential operators. J. Funct. Anal., 257 (2009), 1666–1694. Zbl1179.47041
- W. O. Amrein, D. B. Pearson. Moperators: a generalisation of Weyl–Titchmarsh theory. J. Comp. Appl. Math., 171 (2004), 1–26. Zbl1051.35047
- Yu. M. Arlinskiĭ, E. R. Tsekanovskiĭ. On von Neumann’s problem in extension theory of nonnegative operators. Proc. Amer. Math. Soc., 131 (2003), 3143–3154. Zbl1035.47006
- Yu. M. Arlinskiĭ, E. R. Tsekanovskiĭ. The von Neumann problem for nonnegative symmetric operators. Integr. Eq. Operator Th., 51 (2005), 319–356. Zbl1082.47018
- S. A. Avdonin, M. I. Belishev, S. A. Ivanov. Boundary control and a matrix inverse problem for the equationutt − uxx + V(x)u = 0. Math. USSR Sbornik, 72 (1992), 287–310. Zbl0782.93054
- S. Avdonin, P. Kurasov. Inverse problems for quantum trees. Inverse Probl. Imaging, 2 (2008), 1–21. Zbl1148.35356
- S. Avdonin, S. Lenhart, V. Protopopescu. Solving the dynamical inverse problem for the Schrödinger equation by the boundary control method. Inverse Probl., 18 (2002), 349–361. Zbl1001.35121
- S. Avdonin, S. Lenhart, V. Protopopescu. Determining the potential in the Schrödinger equation from the Dirichlet to Neumann map by the boundary control method. J. Inv. Ill-Posed Probl., 13 (2005), 1–14. Zbl1095.35061
- J. Behrndt, M. Langer. Boundary value problems for partial differential operators on bounded domains. J. Funct. Anal., 243 (2007), 536–565. Zbl1132.47038
- J. Behrndt, M. M. Malamud, H. Neidhardt. Scattering matrices and Weyl functions. Proc. London Math. Soc., 97 (2008), No. 3, 568–598. Zbl1161.47008
- J. F. Brasche, M. M. Malamud, H. Neidhardt.Weyl functions and singular continuous spectra of self-adjoint extensions in Stochastic processes, physics and geometry: New interplays. II. A volume in honor of Sergio Albeverio. F. Gesztesy, H. Holden, J. Jost, S. Paycha, M. Röckner, S. Scarlatti (eds.). Canadian Mathematical Society Conference Proceedings, Vol. 29, Amer. Math. Soc., Providence, RI, 2000, pp. 75–84. Zbl0988.47003
- J. F. Brasche, M. M. Malamud, H. Neidhardt. Weyl function and spectral properties of self-adjoint extensions. Integral Eq. Operator Th., 43 (2002), 264–289. Zbl1008.47028
- 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. Zbl1167.47057
- M. Brown, J. Hinchcliffe, M. Marletta, S. Naboko, I. Wood. The abstract Titchmarsh–Weyl M-function for adjoint operator pairs and its relation to the spectrum. Integral Equ. Operator Th., 63 (2009), 297–320. Zbl1188.47004
- B. M. Brown, M. Marletta. Spectral inclusion and spectral exactness for PDE’s on exterior domains. IMA J. Numer. Anal., 24 (2004), 21–43. Zbl1057.65079
- B. M. Brown, M. Marletta, S. Naboko, I. Wood. Boundary triplets and M-functions for non-selfadjoint operators, with applications to elliptic PDEs and block operator matrices. J. London Math. Soc., 77 (2008), No. 2, 700–718. Zbl1148.35053
- J. Brüning, V. Geyler, K. Pankrashkin. Spectra of self-adjoint extensions and applications to solvable Schrödinger operators. Rev. Math. Phys., 20 (2008), 1–70. Zbl1163.81007
- R. Carmona, J. Lacroix. Spectral theory of random Schrödinger operators. Birkhäuser, Basel, 1990. Zbl0717.60074
- S. Clark, F. Gesztesy. Weyl–Titchmarsh M-function asymptotics and Borg-type theorems for Dirac operators. Trans. Amer. Math. Soc., 354 (2002), 3475–3534. Zbl1017.34019
- S. Clark, F. Gesztesy.On self-adjoint and J-self-adjoint Dirac-type operators: A case study, in Recent advances in differential equations and mathematical physics. N. Chernov, Y. Karpeshina, I. W. Knowles, R. T. Lewis, R. Weikard (eds.). Contemp. Math., Vol. 412, Amer. Math. Soc., Providence, RI, 2006, pp. 103–140. Zbl1124.34062
- E. A. Coddington, N. Levinson. Theory of ordinary differential equations. Krieger, Malabar, 1985. Zbl0064.33002
- V. A. Derkach, S. Hassi, M. M. Malamud, H. S. V. de Snoo. Generalized resolvents of symmetric operators and admissibility. Meth. Funct. Anal. Top., 6 (2000), No.3, 24–55. Zbl0973.47020
- V. Derkach, S. Hassi, M. Malamud, H. de Snoo. Boundary relations and their Weyl families. Trans. Amer. Math. Soc., 358 (2006), 5351–5400. Zbl1123.47004
- V. A. Derkach, M. M. Malamud. Generalized resolvents and the boundary value problems for Hermitian operators with gaps. J. Funct. Anal., 95 (1991), 1–95. Zbl0748.47004
- V. A. Derkach, M. M. Malamud. Characteristic functions of almost solvable extensions of Hermitian operators. Ukrain. Math. J., 44 (1992), 379–401. Zbl0804.47009
- V. A. Derkach, M. M. Malamud. The extension theory of Hermitian operators and the moment problem. J. Math. Sci., 73 (1995), 141–242. Zbl0848.47004
- N. Dunford, J. T. Schwartz. Linear operators Part II: Spectral theory. Interscience, New York, 1988. Zbl0635.47002
- C. Fox, V. Oleinik, B. Pavlov.A Dirichlet-to-Neumann map approach to resonance gaps and bands of periodic networks, in Recent advances in differential equations and mathematical physics. N. Chernov, Y. Karpeshina, I. W. Knowles, R. T. Lewis, R. Weikard (eds.). Contemp. Math. Vol. 412, Amer. Math. Soc., Providence, RI, 2006, pp. 151–169. Zbl1114.34066
- F. Gesztesy, H. Holden. Soliton equations and their algebro-geometric solutions. Volume I: (1 + 1)-Dimensional continuous models. Cambridge Studies in Advanced Mathematics, Vol. 79, Cambridge University Press, Cambridge, 2003. Zbl1061.37056
- F. Gesztesy, H. Holden, B. Simon, Z. Zhao. Higher order trace relations for Schrödinger operators. Rev. Math. Phys., 7 (1995), 893–922. Zbl0833.34084
- F. Gesztesy, N. J. Kalton, K. A. Makarov, E. Tsekanovskii.Some applications of operator-valued Herglotz functions, in Operator theory, system theory and telated topics. The Moshe Livšic anniversary volume. D. Alpay, V. Vinnikov (eds.). Operator Theory: Advances and Applications, Vol. 123, Birkhäuser, Basel, 2001, pp. 271–321. Zbl0991.30020
- F. Gesztesy, K. A. Makarov, E. Tsekanovskii. An Addendum to Krein’s formula. J. Math. Anal. Appl., 222 (1998), 594–606. Zbl0922.47006
- F. Gesztesy, 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: A volume in honor of Vladimir G. Maz’ya’s 70th birthday. D. Mitrea, M. Mitrea (eds.). Proceedings of Symposia in Pure Mathematics, Vol. 79, Amer. Math. Soc., Providence, RI, 2008, pp. 105–173. Zbl1178.35147
- F. Gesztesy, M. Mitrea.Robin-to-Robin maps and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains, in Modern analysis and applications. The Mark Krein cetenary conference, Vol. 2. V. Adamyan, Y. M. Berezansky, I. Gohberg, M. L. Gorbachuk, V. Gorbachuk, A. N. Kochubei, H. Langer, G. Popov (eds.). Operator Theory: Advances and Applications, Vol. 191, Birkhäuser, Basel, 2009, pp. 81–113. Zbl1180.35186
- F. Gesztesy, M. Mitrea. Nonlocal Robin Laplacians and some remarks on a paper by Filonov on eigenvalue inequalities. J. Diff. Eq., 247 (2009), 2871–2896. Zbl1181.35155
- F. Gesztesy, M. Mitrea.Self-adjoint extensions of the Laplacian and Krein-type resolvent formulas in nonsmooth domains. Preprint, 2009. Zbl1231.47044
- F. Gesztesy, M. Mitrea, M. Zinchenko. Variations on a theme of Jost and Pais. J. Funct. Anal., 253 (2007), 399–448. Zbl1133.47010
- F. Gesztesy, R. Ratnaseelan, G. Teschl.The KdV hierarchy and associated trace formulas, in Recent developments in operator theory and its applications. I. Gohberg, P. Lancaster, and P. N. Shivakumar (eds.). Operator Theory: Advances and Applications, Vol. 87, Birkhäuser, Basel, 1996, pp. 125–163. Zbl0865.35116
- F. Gesztesy, B. Simon. Uniqueness theorems in inverse spectral theory for one-dimensional Schrödinger operators. Trans. Amer. Math. Soc., 348 (1996), 349–373. Zbl0846.34090
- F. Gesztesy, E. Tsekanovskii. On matrix-valued Herglotz functions. Math. Nachr., 218 (2000), 61–138. Zbl0961.30027
- F. Gesztesy and M. Zinchenko, Boundary Data Maps, Perturbation Determinants, and Krein-Type Resolvent Formulas for Schrödinger Operators on Compact Intervals, preprint, 2010.
- V. I. Gorbachuk, M. L. Gorbachuk. Boundary value problems for operator differential equations. Kluwer, Dordrecht, 1991. Zbl0751.47025
- G. Grubb. Krein resolvent formulas for elliptic boundary problems in nonsmooth domains. Rend. Semin. Mat. Univ. Politec. Torino, 66 (2008), 271–297. Zbl1206.35090
- G. Grubb. Distributions and operators. Graduate Texts in Mathematics, Vol. 252, Springer, New York, 2009. Zbl1171.47001
- T. Kato. Perturbation theory for linear operators. Corr. printing of the 2nd ed., Springer, Berlin, 1980. Zbl0435.47001
- A. Kiselev, B. Simon. Rank one perturbations with infinitesimal coupling. J. Funct. Anal., 130 (1995), 345–356. Zbl0823.47015
- M. G. Krein, I. E. Ovcharenko. Q-functions and sc-resolvents of nondensely defined hermitian contractions. Sib. Math. J., 18 (1977), 728–746. Zbl0409.47013
- M. G. Krein, I. E. Ovčarenko. Inverse problems for Q-functions and resolvent matrices of positive hermitian operators. Sov. Math. Dokl., 19 (1978), 1131–1134.
- M. G. Krein, S. N. Saakjan. Some new results in the theory of resolvents of hermitian operators. Sov. Math. Dokl., 7 (1966), 1086–1089.
- M. G. Krein, Ju. L. Smul’jan. On linear-fractional transformations with operator coefficients. Amer. Math. Soc. Transl., 103 (1974), No. 2, 125–152.
- P. Kurasov. Triplet extensions I: Semibounded operators in the scale of Hilbert spaces. J. Analyse Math., 107 (2009), 251–286. Zbl1190.47018
- P. Kurasov, S. T. Kuroda. Krein’s resolvent formula and perturbation theory. J. Operator Th., 51 (2004), 321–334. Zbl1070.47008
- H. Langer, B. Textorius. On generalized resolvents and Q-functions of symmetric linear relations (subspaces) in Hilbert space. Pacific J. Math., 72 (1977), 135–165. Zbl0335.47014
- B. M. Levitan. Inverse Sturm–Liouville problems. VNU Science Press, Utrecht, 1987. Zbl0749.34001
- B. M. Levitan, I. S. Sargsjan. Introduction to spectral theory. Amer. Math. Soc., Providence, RI, 1975. Zbl0302.47036
- M. M. Malamud, V. I. Mogilevskii. Krein type formula for canonical resolvents of dual pairs of linear relations. Methods Funct. Anal. Topology, 8 (2002), No. 4, 72–100. Zbl1074.47501
- V. A. Marchenko. Some questions in the theory of one-dimensional linear differential operators of the second order, I. Trudy Moskov. Mat. Obšč., 1 (1952), 327–420. (Russian.) English transl. in Amer. Math. Soc. Transl., Ser. 2, 101 (1973), 1–104.
- V. A. Marchenko. Sturm–Liouville operators and applications. Birkhäuser, Basel, 1986. Zbl0592.34011
- M. Marletta. Eigenvalue problems on exterior domains and Dirichlet to Neumann maps. J. Comp. Appl. Math., 171 (2004), 367–391. Zbl1055.65093
- S. Nakamura. A remark on the Dirichlet–Neumann decoupling and the integrated density of states. J. Funct. Anal., 179 (2001), 136–152. Zbl0970.35084
- G. Nenciu. Applications of the Kreĭn resolvent formula to the theory of self-adjoint extensions of positive symmetric operators. J. Operator Th., 10 (1983), 209–218. Zbl0561.47005
- K. Pankrashkin. Resolvents of self-adjoint extensions with mixed boundary conditions. Rep. Math. Phys., 58 (2006), 207–221. Zbl1143.47017
- B. Pavlov. The theory of extensions and explicitly-soluble models. Russ. Math. Surv., 42:6 (1987), 127–168. Zbl0665.47004
- B. Pavlov.S-matrix and Dirichlet-to-Neumann operators. Ch. 6.1.6 in Scattering: Scattering and inverse scattering in pure and applied science, Vol. 2. R. Pike, P. Sabatier (eds.). Academic Press, San Diego, 2002, pp. 1678–1688.
- B. Pavlov. Krein formula with compensated singularities for the ND-mapping and the generalized Kirchhoff condition at the Neumann Schrödinger junction. Russ. J. Math. Phys., 15 (2008), 364–388. Zbl1181.81054
- D. B. Pearson. Quantum scattering and spectral theory. Academic Press, London, 1988. Zbl0673.47011
- A. Posilicano. A Krein-like formula for singular perturbations of self-adjoint operators and applications. J. Funct. Anal., 183 (2001), 109–147. Zbl0981.47022
- A. Posilicano. Self-adjoint extensions by additive perturbations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), 2 (2003), No. 1, 1–20. Zbl1096.47505
- A. Posilicano. Boundary triples and Weyl functions for singular perturbations of self-adjoint operators. Meth. Funct. Anal. Topology, 10 (2004), No.2, 57–63. Zbl1066.47024
- A. Posilicano. Self-adjoint extensions of restrictions. Operators and Matrices, 2 (2008), 483–506. Zbl1175.47025
- A. Posilicano, L. Raimondi.Krein’s resolvent formula for self-adjoint extensions of symmetric second-order elliptic differential operators. J. Phys. A: Math. Theor., 42 (2009), 015204 (11pp). Zbl1161.81016
- A. Rybkin. On the boundary control approach to inverse spectral and scattering theory for Schrödinger operators. Inverse Probl. Imaging, 3 (2009), 139–149. Zbl1185.35332
- V. Ryzhov. A general boundary value problem and its Weyl function. Opuscula Math., 27 (2007), 305–331. Zbl1155.47025
- V. Ryzhov. Weyl–Titchmarsh function of an abstract boundary value problem, operator colligations, and linear systems with boundary control. Complex Anal. Operator Theory, 3 (2009), 289–322. Zbl1182.47011
- V. Ryzhov.Spectral boundary value problems and their linear operators. Preprint, 2009.
- Sh. N. Saakjan. On the theory of the resolvents of a symmetric operator with infinite deficiency indices. Dokl. Akad. Nauk Arm. SSR, 44 (1965), 193–198. (Russian) .
- A. V. Straus. Extensions and generalized resolvents of a non-densely defined symmetric operator. Math. USSR Izv., 4 (1970), 179–208.
- E. C. Titchmarsh. Eigenfunction expansions, Part I. 2nd ed., Clarendon Press, Oxford, 1962. Zbl0099.05201
- E. C. Titchmarsh. The theory of functions. 2nd ed., Oxford University Press, Oxford, 1985. Zbl0005.21004
- E. R. Tsekanovskii, Yu. L. Shmul’yan. The theory of bi-extensions of operators on rigged Hilbert spaces. Unbounded operator colligations and characteristic functions. Russ. Math. Surv., 32:5 (1977), 73–131.

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