Opuscula Math. 38, no. 5 (2018), 681-698

Opuscula Mathematica

Krein-von Neumann extension of an even order differential operator on a finite interval

Yaroslav I. Granovskyi
Leonid L. Oridoroga

Abstract. We describe the Krein-von Neumann extension of minimal operator associated with the expression \(\mathcal{A}:=(-1)^n\frac{d^{2n}}{dx^{2n}}\) on a finite interval \((a,b)\) in terms of boundary conditions. All non-negative extensions of the operator \(A\) as well as extensions with a finite number of negative squares are described.

Keywords: non-negative extension, Friedrichs' extension, Krein-von Neumann extension, boundary triple, Weyl function.

Mathematics Subject Classification: 47A05.

Full text (pdf)

  1. N.I. Akhiezer, I.M. Glazman, Theory of Linear Operators in Hilbert Spaces, Nauka, Moscow, 1978.
  2. A.Yu. Ananieva, V.S. Budyika, To the spectral theory of the Bessel operator on finite interval and half-line, J. of Math. Scien. 211 (2015) 5, 624-645.
  3. T. Ando, K. Nishio, Positive selfadjoint extensions of positive symmetric operators, Tohoku Math. J. 22 (1970), 65-75.
  4. M.S. Ashbaugh, F. Gesztesy, M. Mitrea, R. Shterenberg, G. Teschl, Spectral theory for perturbed Krein Laplacians in nonsmooth domains, Adv. Math. 223 (2010), 1372-1467.
  5. M.S. Ashbaugh, F. Gesztesy, M. Mitrea, R. Shterenberg, G. Teschl, The Krein-von Neumann extension and its connection to an abstract buckling problem, Math. Nachr. 283 (2010) 2, 165-179.
  6. M.S. Ashbaugh, F. Gesztesy, M. Mitrea, R. Shterenberg, G. Teschl, A survey of the Krein-von Neumann extension, the corresponding abstract buckling problem, and Weyl-type spectral asymptotics for perturbed Krein Laplacians in non smooth domains, [in:] M. Demuth and W. Kirsh (eds.), Mathematical Physics, Spectral Theory and Stochastic Analysys, Operator Theory: Advances and Applications 232, Birkhäuser, Springer, Basel (2013), 1-106.
  7. J. Behrndt, F. Gesztesy, T. Micheler, M. Mitrea, The Krein-von Neumann realization of perturbed Laplacians on bounded Lipschitz domains, Operator Theory: Advances and Applications 255, Birkhäuser, Springer, Basel (2016), 49-66.
  8. J. Behrndt, T. Micheler, Elliptic differential operators on Lipschitz domains and abstract boundary value problems, J. Funct. Anal. 267 (2014), 3657-3709.
  9. M.Sh. Birman, Perturbations of the continuous spectrum of a singular elliptic operator by varying the boundary and the boundary conditions, Vestnik Leningrad Univ. 17 (1962) 1, 22-55 [in Russian]; transl. in Spectral Theory of Differential Operators: M. Sh. Birman 80th Anniversary Collection, T. Suslina and D. Yafaev (eds.), AMS Translation, Ser. 2, Advances in the Mathematical Sciences 225, Amer. Math. Soc., Providence, RI (2008), 19-53.
  10. B.M. Brown, J.S. Christiansen, On the Krein and Friedrichs extension of a positive Jacobi operator, Expo. Math. 23 (2005), 176-186.
  11. L. Bruneau, J. Dereziński, V. Georgescu, Homogeneous Schrödinger Operators on Half-line, Ann. Henri Poincaré 12 (2011), 547-590.
  12. V.A. Derkach, M.M. Malamud, Generalized rezolvent and the boundary value problems for Hermitian operators with gaps, J. Funct. Anal. 95 (1991) 1, 1-95.
  13. V.A. Derkach, M.M. Malamud, Characteristic of almost solvable extensions of a Hermitian operators, Ukr. Mat. Zh. 44 (1992) 4, 435-459.
  14. V.A. Derkach, M.M. Malamud, The extension theory of Hermitian operators and the moment problem, J. Math. Sci. (New York) 73 (1995), 141-242.
  15. V.A. Derkach, M.M. Malamud, Extension theory of symmetric operators and boundary value problems, Proceedings of Institute of Mathematics of NAS of Ukraine 104 (2017) [in Russian].
  16. J. Eckhardt, F. Gesztesy, R. Nichols, G. Teschl, Weyl-Titchmarsh Theory for Sturm-Liouville Operators With Distributional Potentials, Opuscula Math. 33 (2013) 3, 467-563.
  17. W.N. Everitt, H. Kalf, The Bessel differential equation and the Hankel transform, Jour. of Comput. and App. Math. 208 (2007), 3-19.
  18. H. Freudental, Über die Friedrichsche Fortsetzung halbbeschränkter Hermitescher Operatoren, Kon. Akad. Wetensch., Amsterdam, Proc. 39 (1936), 832-833.
  19. C. Fulton, Titchmarsh-Weyl m-functions for second-order Sturm-Liouville problems with two singular endpoints, Math. Nachr. 281 (2008) 10, 1418-1475.
  20. C. Fulton, H. Langer, Sturm-Liouville operators with singularities and generalized Nevanlinna functions, Complex Analysis and Operator Theory 4 (2010) 2, 179-243.
  21. F. Gesztesy, M. Mitrea, A description of all self-adjoint extensions of the laplacian and Krein-type resolvent formulas on non-smooth domains, J. Analyse Math. 113 (2011), 53-172.
  22. F. Gesztesy, M. Mitrea, Robin-to-Robin maps and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains, [in:] V. Adamyan, Y.M. Berezansky, I. Gohberg, M.L. Gorbachuk, V. Gorbachuk, A.N. Kochubei, H. Langer, and G. Popov (eds.), Modern Analysis and Applications. The Mark Krein Centenary Conference 2, Operator Theory: Advances and Applications, vol. 191, Birkhäuser, Basel, 2009, 81-113.
  23. V.I. Gorbachuk, M.L. Gorbachuk, Boundary Value Problems for Operator Differential Equations, Mathematics and its Applications (Soviet Series), vol. 48, Kluwer Academic Publishers Group, Dordrecht, 1991.
  24. G. Grubb, A characterization of the non-local boundary value problems associated with an elliptic operator, Ann. Scuola Norm. Sup. Pisa 22 (1968) 3, 425-513.
  25. G. Grubb, Spectral asymptotics for the "soft" selfadjoint extension of a symmetric elliptic differential operator, J. Operator Th. 10 (1983), 9-20.
  26. S. Hassi, M. Malamud, H. de Snoo, On Krein's extension theory of nonnegative operators, Math. Nachr. 274-275 (2004), 40-73.
  27. H. Kalf, A characterization of the Friedrichs extension of Sturm-Liouville operators, J. London Math. Soc. 17 (1978) 2, 511-521.
  28. T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, Heidelberg, New York, 1966.
  29. M.G. Krein, The theory of self-adjoint extensions of semibounded Hermitian transformations and its applications, I, Sb. Math. 20 (1947) 3, 431-495; II, ibid., 21 3, 365-404 [in Russian].
  30. A.A. Lunyov, Spectral functions of the simplest even order ordinary differential operator, Methods of Functional Analysis and Topology 19 (2013) 4, 319-326.
  31. M.M. Malamud, Spectral theory of elliptic operators in exterior domains, Russ. J. Math. Phys. 17 (2010), 96-125.
  32. M.M. Malamud, H. Neidhardt, Sturm-Liouville boundary value problems with operator potentials and unitary equivalence, J. Differential Equations 252 (2012), 5875-5922.
  33. J. von Neumann, Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren, Math. Ann. 102 (1929), 49-131.
  34. F.S. Rofe-Beketov, Self-adjoint extensions of differential operators in a space of vector-valued functions, Teor. Funkcii Funkcional. Anal. i Prilozen. 8 (1969), 3-24 [in Russian].
  • Yaroslav I. Granovskyi
  • NAS of Ukraine, Institute of Applied Mathematics and Mechanics, Ukraine
  • Leonid L. Oridoroga
  • Donetsk National University, Donetsk, Ukraine
  • Communicated by A.A. Shkalikov.
  • Received: 2017-08-21.
  • Revised: 2017-12-15.
  • Accepted: 2017-12-20.
  • Published online: 2018-06-12.
Opuscula Mathematica - cover

Cite this article as:
Yaroslav I. Granovskyi, Leonid L. Oridoroga, Krein-von Neumann extension of an even order differential operator on a finite interval, Opuscula Math. 38, no. 5 (2018), 681-698, https://doi.org/10.7494/OpMath.2018.38.5.681

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.