Functional Model of Dissipative Fourth Order Differential Operators

Tuna, Hüseyin

doi:10.2478/auom-2013-0036

Abstract

In this paper, maximal dissipative fourth order operators with equal deficiency indices are investigated. We construct a self adjoint dilation of such operators. We also construct a functional model of the maximal dissipative operator which based on the method of Pavlov and define its characteristic function. We prove theorems on the completeness of the system of eigenvalues and eigenvectors of the maximal dissipative fourth order operators.

References

[1] B.P. Allahverdiev, On dilation theory and spectral analysis of dissipative Schrödinger operators in Weyl’s limit-circle case (Russian), Izv. Akad. Nauk. SSSR Ser. Mat.54 (1990), 242 - 257; English transl.: Math. USSR Izv. 36 (1991), 247 - 262.
Search in Google Scholar
[2] B.P. Allahverdiev, Spectral analysis of nonselfadjoint Schrödinger operators with a matrix potential, J. Math. Anal. Appl. 303 (2005), 208 - 219.10.1016/j.jmaa.2004.08.031
Search in Google Scholar
[3] B.P. Allahverdiev, A nonself-adjoint singular Sturm-Liouville problem with a spectral parameter in the boundary condition, Math. Nachr. 278, 7 - 8 (2005), 743 - 755.
Search in Google Scholar
[4] B.P. Allahverdiev, Spectral analysis of dissipative Dirac operators with general boundary conditions, J. Math. Anal. Appl. 283 (2003), 287 - 303.10.1016/S0022-247X(03)00293-2
Search in Google Scholar
[5] B.P. Allahverdiev, Dilation and Functional Model of Dissipative Operator Generated by an Infinite Jacobi Matrix, Math. and Comp. Modelling 38, 3 (2003), 989 - 1001.10.1016/S0895-7177(03)90101-4
Search in Google Scholar
[6] M. Baro, H. Neidhardt Dissipative Schrodinger-type operators as a model for generation and recombination, J. Math. Phys. 44, 6 (2003), 2373 - 2401.10.1063/1.1562748
Search in Google Scholar
[7] M. Baro, H.-Chr. Kaiser, H. Neidhardt, J. Rehberg, Dissipative Schrödinger-Poisson systems, J. Math. Phys. 45,1 (2004), 21 - 43.10.1063/1.1628385
Search in Google Scholar
[8] M. Baro, H.-Chr. Kaiser, H. Neidhardt, J. Rehberg, A Quantum Transmitting Schrodinger-Poisson System, Rev. Math. Phys. 16, 3 (2004), 281 - 330.10.1142/S0129055X04001996
Search in Google Scholar
[9] J. Behrndt, M.M. Malamud, H. Neidhardt: Scattering theory for open quantum systems with finite rank coupling, Math. Phys. Anal. Geom. 10, (2007), 313 - 358.10.1007/s11040-008-9035-x
Search in Google Scholar
[10] J. Behrndt, M.M. Malamud, H. Neidhardt: Scattering matrices and Weyl functions, Proc. London Math. Soc. 97, (2008), 568 - 598.10.1112/plms/pdn016
Search in Google Scholar
[11] J. Behrndt, M.M. Malamud, H. Neidhardt: Trace formulae for dissipative and coupled scattering systems, Operator Theory Advances Applications 188, (2008), 49 - 87.10.1007/978-3-7643-8911-6_4
Search in Google Scholar
[12] J. Behrndt, M.M. Malamud, H. Neidhardt: Finite rank perturbations, scattering matrices and inverse problems, Operator Theory Advances Applications 198, (2009) , 61 - 85.
Search in Google Scholar
[13] C. T. Fulton, The Bessel-squared equation in the lim-2, lim-3, and lim-4 cases, Quart. J. Math. Oxford (2), 40, (1989), 423 - 456.10.1093/qmath/40.4.423
Search in Google Scholar
[14] M. L. Gorbachuk and V.I. Gorbachuk, Boundary Value Problems for Operator Differential Equations, Naukova Dumka, Kiev, 1984; English transl. 1991,Birkhauser Verlag.10.1007/978-94-011-3714-0
Search in Google Scholar
[15] A. N. Kochubei, Extensions of symmetric operators and symmetric binary relations, Mat. Zametki 17, (1975), 41 - 48; English transl. in Math. Notes 17 (1975), 25 - 28.
Search in Google Scholar
[16] A. Kuzhel, Characteristic Functions and Models of Nonselfadjoint Operators, Kluwer Academic, Dordrecht, 1996.10.1007/978-94-009-0183-4
Search in Google Scholar
[17] S. Kuzhel, On the inverse problem in the Lax-Phillips scattering theory method for a class of operator-differential equations, St. Petersburg Math. J. 13 (2002), 41 - 56.
Search in Google Scholar
[18] P. D. Lax and R.S. Phillips, Scattering Theory, Academic Press, New York, 1967.
Search in Google Scholar
[19] B. Sz. Nagy and C., Foias, Analyse Harmonique des Operateurs de L’espace de Hilbert, Masson, Akad. Kiado, Paris, Budapest, 1967, English transl. North-Holland, Amsterdam, and Akad. Kiado, Budapest, 1970.
Search in Google Scholar
[20] M. A. Naimark, Linear Differential Operators, 2nd edn., 1968, Nauka, Moscow, English transl. of 1st. edn., 1, 2, 1969, New York.
Search in Google Scholar
[21] B. S. Pavlov, Selfadjoint Dilation of a Dissipative Schrödinger Operator and Eigenfunction Expansion, Funct. Anal. Appl., vol. 98, (1975), 172 - 173.10.1007/BF01075465
Search in Google Scholar
[22] B. S. Pavlov, Selfadjoint Dilation of a Dissipative Schrödinger Operator and its Resolution in terms of Eigenfunctions, Math. USSR Sbornik, vol. 31, no.4, (1977), 457 - 478.10.1070/SM1977v031n04ABEH003716
Search in Google Scholar
[23] B. S. Pavlov, Dilation theory and spectral analysis of nonselfadjoint differential operators, Proc. 7th Winter School, Drobobych (1974), 3 - 69, (1976), (Russian); English transl: Transl. II. Ser., Am. Math. Soc. 115, (1981), 103 - 142.
Search in Google Scholar

Functional Model of Dissipative Fourth Order Differential Operators

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