Friedrichs extension of operators defined by linear Hamiltonian systems on unbounded interval

Roman Šimon Hilscher; Petr Zemánek

Mathematica Bohemica (2010)

  • Volume: 135, Issue: 2, page 209-222
  • ISSN: 0862-7959

Abstract

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In this paper we consider a linear operator on an unbounded interval associated with a matrix linear Hamiltonian system. We characterize its Friedrichs extension in terms of the recessive system of solutions at infinity. This generalizes a similar result obtained by Marletta and Zettl for linear operators defined by even order Sturm-Liouville differential equations.

How to cite

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Hilscher, Roman Šimon, and Zemánek, Petr. "Friedrichs extension of operators defined by linear Hamiltonian systems on unbounded interval." Mathematica Bohemica 135.2 (2010): 209-222. <http://eudml.org/doc/38125>.

@article{Hilscher2010,
abstract = {In this paper we consider a linear operator on an unbounded interval associated with a matrix linear Hamiltonian system. We characterize its Friedrichs extension in terms of the recessive system of solutions at infinity. This generalizes a similar result obtained by Marletta and Zettl for linear operators defined by even order Sturm-Liouville differential equations.},
author = {Hilscher, Roman Šimon, Zemánek, Petr},
journal = {Mathematica Bohemica},
keywords = {linear Hamiltonian system; Friedrichs extension; self-adjoint operator; recessive solution; quadratic functional; positivity conjoined basis; linear Hamiltonian system; Friedrichs extension; selfadjoint operator; recessive solution; quadratic functional; positivity conjoined basis},
language = {eng},
number = {2},
pages = {209-222},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Friedrichs extension of operators defined by linear Hamiltonian systems on unbounded interval},
url = {http://eudml.org/doc/38125},
volume = {135},
year = {2010},
}

TY - JOUR
AU - Hilscher, Roman Šimon
AU - Zemánek, Petr
TI - Friedrichs extension of operators defined by linear Hamiltonian systems on unbounded interval
JO - Mathematica Bohemica
PY - 2010
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 135
IS - 2
SP - 209
EP - 222
AB - In this paper we consider a linear operator on an unbounded interval associated with a matrix linear Hamiltonian system. We characterize its Friedrichs extension in terms of the recessive system of solutions at infinity. This generalizes a similar result obtained by Marletta and Zettl for linear operators defined by even order Sturm-Liouville differential equations.
LA - eng
KW - linear Hamiltonian system; Friedrichs extension; self-adjoint operator; recessive solution; quadratic functional; positivity conjoined basis; linear Hamiltonian system; Friedrichs extension; selfadjoint operator; recessive solution; quadratic functional; positivity conjoined basis
UR - http://eudml.org/doc/38125
ER -

References

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  1. Ahlbrandt, C. D., 10.1216/RMJ-1972-2-2-169, Rocky Mountain J. Math. 2 (1972), 169-182. (1972) MR0296388DOI10.1216/RMJ-1972-2-2-169
  2. Baxley, J. V., The Friedrichs extension of certain singular differential operators, Duke Math. J. 35 (1968), 455-462. (1968) Zbl0174.45701MR0226446
  3. Brown, B. M., Christiansen, J. S., 10.1016/j.exmath.2005.01.020, Expo. Math. 23 (2005), 179-186. (2005) Zbl1078.39019MR2155010DOI10.1016/j.exmath.2005.01.020
  4. Došlý, O., Principal and nonprincipal solutions of symplectic dynamic systems on time scales, Electron. J. Qual. Theory Differ. Equ. 2000, Suppl. No. 5, 14 p., electronic only. MR1798655
  5. Došlý, O., Hasil, P., Friedrichs extension of operators defined by symmetric banded matrices, Linear Algebra Appl. 430 (2009), 1966-1975. (2009) Zbl1171.39004MR2503945
  6. Freudenthal, H., Über die Friedrichssche Fortsetzung halbbeschränkter Hermitescher Operatoren, {German} Proc. Akad. Wet. Amsterdam 39 (1936), 832-833. (1936) Zbl0015.25904
  7. Friedrichs, K., 10.1007/BF01449150, German Math. Ann. 109 (1934), 465-487. (1934) Zbl0009.07205MR1512905DOI10.1007/BF01449150
  8. Friedrichs, K., 10.1007/BF01449164, German Math. Ann. 109 (1934), 685-713. (1934) MR1512919DOI10.1007/BF01449164
  9. Friedrichs, K., 10.1007/BF01565401, German Math. Ann. 112 (1936), 1-23. (1936) MR1513033DOI10.1007/BF01565401
  10. Kalf, H., 10.1112/jlms/s2-17.3.511, J. London Math. Soc. 17 (1978), 511-521. (1978) Zbl0406.34029MR0492493DOI10.1112/jlms/s2-17.3.511
  11. Kratz, W., Quadratic Functionals in Variational Analysis and Control Theory, Akademie Verlag, Berlin (1995). (1995) Zbl0842.49001MR1334092
  12. Marletta, M., Zettl, A., 10.1006/jdeq.1999.3685, J. Differential Equations 160 (2000), 404-421. (2000) Zbl0954.47012MR1736997DOI10.1006/jdeq.1999.3685
  13. Möller, M., Zettl, A., 10.1006/jdeq.1995.1003, J. Differential Equations 115 (1995), 50-69. (1995) MR1308604DOI10.1006/jdeq.1995.1003
  14. Niessen, H. D., Zettl, A., The Friedrichs extension of regular ordinary differential operators, Proc. Roy. Soc. Edinburgh Sect. A 114 (1990), 229-236. (1990) Zbl0712.34020MR1055546
  15. Niessen, H. D., Zettl, A., Singular Sturm-Liouville problems: the Friedrichs extension and comparison of eigenvalues, Proc. London Math. Soc. 64 (1992), 545-578. (1992) Zbl0768.34015MR1152997
  16. Reid, W. T., Ordinary Differential Equations, Wiley, New York (1971). (1971) Zbl0212.10901MR0273082
  17. Rellich, F., 10.1007/BF01342848, German Math. Ann. 122 (1951), 343-368. (1951) Zbl0044.31201MR0043316DOI10.1007/BF01342848
  18. Rosenberger, R., 10.1112/jlms/s2-31.3.501, J. London Math. Soc. 31 (1985), 501-510. (1985) Zbl0615.34019MR0812779DOI10.1112/jlms/s2-31.3.501
  19. Wang, A., Sun, J., Zettl, A., 10.1016/j.jde.2008.11.001, J. Differential Equations 246 (2009), 1600-1622. (2009) Zbl1169.47033MR2488698DOI10.1016/j.jde.2008.11.001
  20. Zettl, A., On the Friedrichs extension of singular differential operators, Commun. Appl. Anal. 2 (1998), 31-36. (1998) Zbl0895.34018MR1612893
  21. Zettl, A., Sturm-Liouville Theory, Mathematical Surveys and Monographs, Vol. 121, American Mathematical Society, Providence, RI (2005). (2005) Zbl1103.34001MR2170950
  22. Zheng, Z., Chen, S., 10.1016/j.amc.2006.05.041, Appl. Math. Comput. 182 (2006), 1514-1527. (2006) Zbl1118.47038MR2282593DOI10.1016/j.amc.2006.05.041
  23. Zheng, Z., Qi, J., Chen, S., 10.1016/j.camwa.2008.05.043, Comput. Math. Appl. 56 (2008), 2825-2833. (2008) Zbl1165.34425MR2467672DOI10.1016/j.camwa.2008.05.043

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