Abstract
We introduce GBDT version of Darboux transformation for Hamiltonian and Shin–Zettl systems as well as for Sturm–Liouville equations (including indefinite Sturm–Liouville equations). These are the first results on Darboux transformation for general-type Hamiltonian and for Shin–Zettl systems. The obtained results are applied to the corresponding transformations of the Weyl–Titchmarsh functions and to the construction of explicit solutions of dynamical systems, of two-way diffusion equations and of indefinite Sturm–Liouville equations. The energy of the explicit solutions of dynamical systems is expressed (in a quite simple form) in terms of the parameter matrices of GBDT. The insertion of non-real eigenvalues into the spectrum of indefinite Sturm–Liouville operators is studied.
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References
Amrein, W.O., Hinz, A.M., Pearson, D.B. (eds.): Sturm–Liouville Theory. Past and Present. Birkhäuser, Basel (2005)
Atkinson, F.V.: Discrete and Continuous Boundary Problems. Mathematics in Science and Engineering, vol. 8. Academic Press, London (1964)
Atkinson, F.V., Everitt, W.N., Ong, K.S.: On the \(m\)-coefficient of Weyl for a differential equation with an indefinite weight function. Proc. Lond. Math. Soc. s3–29(2), 368–384 (1974)
Beals, R.: Partial-range completeness and existence of solutions to two-way diffusion equations. J. Math. Phys. 22, 954–960 (1981)
Behncke, H., Hinton, D.B.: Transformation theory of symmetric differential expressions. Adv. Differ. Equ. 11, 601–626 (2006)
Behrndt, J.: An open problem: accumulation of nonreal eigenvalues of indefinite Sturm–Liouville operators. Integral Equ. Oper. Theory 77, 299–301 (2013)
Behrndt, J., Philipp, F., Trunk, C.: Bounds on the non-real spectrum of differential operators with indefinite weights. Math. Ann. 357, 185–213 (2013)
Binding, P., Langer, H., Möller, M.: Oscillation results for Sturm–Liouville problems with an indefinite weight function. J. Comput. Appl. Math. 171, 93–101 (2004)
Cieslinski, J.L.: Algebraic construction of the Darboux matrix revisited. J. Phys. A 42, 404003 (2009). 40 pp
Crum, M.M.: Associated Sturm–Liouville systems. Q. J. Math. Oxf. II Ser. 6, 121–127 (1955)
Curgus, B., Langer, H.: A Krein space approach to symmetric ordinary differential operators with an indefinite weight function. J. Differ. Equ. 79, 31–61 (1989)
Deift, P.A.: Applications of a commutation formula. Duke Math. J. 45, 267–310 (1978)
Eckhardt, J., Gesztesy, F., Nichols, R., Sakhnovich, A.L., Teschl, G.: Inverse spectral problems for Schrödinger-type operators with distributional matrix-valued potentials. Differ. Integral Equ. 28, 505–522 (2015)
Everitt, W.N., Race, D.: Some remarks on linear ordinary quasi-differential expressions. Proc. Lond. Math. Soc. s3–54(2), 300–320 (1987)
Everitt, W.N., Race, D.: The regular representation of singular second-order differential expressions using quasi-derivatives. Proc. Lond. Math. Soc. s3–65(2), 383–404 (1992)
Fisch, N.J., Kruskal, M.D.: Separating variables in two-way diffusion equations. J. Math. Phys. 21, 740–750 (1980)
Gesztesy, F.: A complete spectral characterization of the double commutation method. J. Funct. Anal. 117, 401–446 (1993)
Gesztesy, F., Teschl, G.: On the double commutation method. Proc. Am. Math. Soc. 124, 1831–1840 (1996)
Gohberg, I., Kaashoek, M.A., Sakhnovich, A.L.: Sturm–Liouville systems with rational Weyl functions: explicit formulas and applications. Integral Equ. Oper. Theory 30, 338–377 (1998)
Gohberg, I., Kaashoek, M.A., Sakhnovich, A.L.: Scattering problems for a canonical system with a pseudo-exponential potential. Asymptot. Anal. 29, 1–38 (2002)
Gu, C.H., Hu, H., Zhou, Z.: Darboux Transformations in Integrable Systems. Theory and Their Applications to Geometry, Mathematical Physics Studies, vol. 26. Springer, Berlin (2005)
Hinton, D.B., Shaw, J.K.: On Titchmarsh–Weyl \(M(\lambda )\)-functions for linear Hamiltonian systems. J. Differ. Equ. 40, 316–342 (1981)
Hinton, D.B., Schneider, A.: On the Titchmarsh–Weyl coefficients for singular \(S\)-Hermitian systems II. Math. Nachr. 185, 67–84 (1997)
Kac, I.S., Krein, M.G.: On the spectral functions of the string. Am. Math. Soc. Transl. 2(103), 19–102 (1974)
Jacob, B., Zwart, H.J.: Linear Port-Hamiltonian Systems on Infinite-Dimensional Spaces, Operator Theory: Advances and Applications, vol. 223. Birkhäuser, Basel (2012)
Karabash, I., Trunk, C.: Spectral properties of singular Sturm–Liouville operators with indefinite weight \({\rm sgn}\, x\). Proc. R. Soc. Edinb. Sect. A 139, 483–503 (2009)
Kikonko, M., Mingarelli, A.B.: Bounds on real and imaginary parts of non-real eigenvalues of a non-definite Sturm–Liouville problem. J. Differ. Equ. 261, 6221–6232 (2016)
Kostenko, A.: The similarity problem for indefinite Sturm–Liouville operators and the HELP inequality. Adv. Math. 246, 368–413 (2013)
Kostenko, A., Sakhnovich, A., Teschl, G.: Commutation methods for Schrödinger operators with strongly singular potentials. Math. Nachr. 285, 392–410 (2012)
Krall, A.M.: \(M(\lambda )\) theory for singular Hamiltonian systems with one singular point. SIAM J. Math. Anal. 20, 664–700 (1989)
Krall, A.M.: A limit-point criterion for linear Hamiltonian systems. Appl. Anal. 61, 115–119 (1996)
Krein, M.G.: On a continual analogue of a Christoffel formula from the theory of orthogonal polynomials. Dokl. Akad. Nauk SSSR (N.S.) 113, 970–973 (1957). (Russian)
Langer, M., Woracek, H.: Indefinite Hamiltonian systems whose Titchmarsh–Weyl coefficients have no finite generalized poles of non-positive type. Oper. Matrices 7, 477–555 (2013)
Levitin, M., Seri, M.: Accumulation of complex eigenvalues of an indefinite Sturm–Liouville operator with a shifted Coulomb potential. Oper. Matrices 10, 223–245 (2016)
Matveev, V.B., Salle, M.A.: Darboux Transformations and Solitons. Springer, Berlin (1991)
Mennicken, R., Sakhnovich, A.L., Tretter, C.: Direct and inverse spectral problem for a system of differential equations depending rationally on the spectral parameter. Duke Math. J. 109, 413–449 (2001)
Mirzoev, K.A.: Sturm–Liouville operators. Trans. Mosc. Math. Soc. 75, 281–299 (2014)
Mogilevskii, V.: Spectral and pseudospectral functions of Hamiltonian systems: development of the results by Arov–Dym and Sakhnovich. Methods Funct. Anal. Topol. 21, 370–402 (2015)
Naimark, M.A.: Linear Differ. Oper. Frederick Ungar, New York (1968)
Qi, J., Xie, B., Chen, S.: The upper and lower bounds on non-real eigenvalues of indefinite Sturm–Liouville problems. Proc. Am. Math. Soc. 144, 547–559 (2016)
Richardson, R.: Contributions to the study of oscillation properties of the solutions of linear differential equations of the second order. Am. J. Math. 40, 283–316 (1918)
Sakhnovich, A.L.: Dressing procedure for solutions of nonlinear equations and the method of operator identities. Inverse Prob. 10, 699–710 (1994)
Sakhnovich, A.L.: Iterated Bäcklund–Darboux transform for canonical systems. J. Funct. Anal. 144, 359–370 (1997)
Sakhnovich, A.L.: Generalized Bäcklund–Darboux transformation: spectral properties and nonlinear equations. J. Math. Anal. Appl. 262, 274–306 (2001)
Sakhnovich, A.L.: Dirac type system on the axis: explicit formulas for matrix potentials with singularities and soliton–positon interactions. Inverse Prob. 19, 845–854 (2003)
Sakhnovich, A.L., Sakhnovich, L.A., Roitberg, I.Y.: Inverse Problems and Nonlinear Evolution Equations. Solutions, Darboux Matrices and Weyl-Titchmarsh Functions. De Gruyter Studies in Mathematics, vol. 47. De Gruyter, Berlin (2013)
Sakhnovich, L.A.: On the factorization of the transfer matrix function. Sov. Math. Dokl. 17, 203–207 (1976)
Sakhnovich, L.A.: Spectral Theory of Canonical Differential Systems, Method of Operator Identities. Operator Theory: Advances and Applications, vol. 107. Birkhäuser, Basel (1999)
Schmid, H., Tretter, C.: Singular Dirac systems and Sturm–Liouville problems nonlinear in the spectral parameter. J. Differ. Equ. 181, 511–542 (2002)
Shin, D.: On quasi-differential operators in Hilbert space. Dokl. Akad. Nauk SSSR 18, 523–526 (1938)
Šepitka, P., Simon Hilscher, R.: Genera of conjoined bases of linear Hamiltonian systems and limit characterization of principal solutions at infinity. J. Differ. Equ. 260, 6581–6603 (2016)
Teschl, G.: Deforming the point spectra of one-dimensional Dirac operators. Proc. Am. Math. Soc. 126, 2873–2881 (1998)
Zettl, A.: Formally self-adjoint quasi-differential operators. Rocky Mt. J. Math. 5, 453–474 (1975)
Zettl, A.: Sturm–Liouville Theory. American Mathematical Society, Providence (2005)
Zettl, A., Sun, J.: Survey article: self-adjoint ordinary differential operators and their spectrum. Rocky Mt. J. Math. 45, 763–886 (2015)
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This research was supported by the Austrian Science Fund (FWF) under Grant No. P29177.
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Sakhnovich, A. Hamiltonian Systems and Sturm–Liouville Equations: Darboux Transformation and Applications. Integr. Equ. Oper. Theory 88, 535–557 (2017). https://doi.org/10.1007/s00020-017-2385-7
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DOI: https://doi.org/10.1007/s00020-017-2385-7