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An eigenfunction expansion for a quadratic pencil of a Schrödinger operator with spectral singularities. (English) Zbl 0945.34067

In the previous paper [J. Differ. Equations 151, No. 2, 252-267 (1999; Zbl 0927.34063)], the authors started the study of quadratic pencils of Schrödinger operators on \(L^2(\mathbb{R}_+)\) generated by differential expressions of the form \[ l(y)= -y''+ [q(x)+ 2\lambda p(x)- \lambda^2]y.\tag{1} \] Using part of the results of that paper they obtain now a two-fold spectral expansion of operators \(L\) generated by the differential expression (1) and the boundary condition \(y(0)= 0\). It is assumed that the complex-valued functions \(p\) and \(q\) satisfy the conditions \(\lim_{x\to\infty} p(x)= 0\) and \(\sup_{x\in\mathbb{R}_+}\{e^{\varepsilon x}[|q(x)|+|p'(x)|\}< \infty\) for some \(\varepsilon> 0\). The main step in the proof consists in the regularization of some integrals which diverge in the neighbourhood of spectral singularities. The spectral expansion converges in appropriate weighted \(L^2\)-spaces.

MSC:

34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
47A70 (Generalized) eigenfunction expansions of linear operators; rigged Hilbert spaces

Citations:

Zbl 0927.34063
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References:

[1] Degasperis, A., On the inverse problem for the Klein-Gordon \(s\), J. Math. Phys., 11, 551-567 (1970)
[2] Gasymov, M. G., On the decomposition in a series of eigenfunctions for a non-selfadjoint boundary value problem of the solution of a differential equation with a singularity at zero point, Soviet Math. Dokl., 6, 1426-1429 (1965) · Zbl 0145.32702
[3] Gasymov, M. G., Expansion in terms of the solutions of a scattering theory problem for the non-selfadjoint Schrödinger equation, Soviet Math. Dokl., 9, 390-393 (1968) · Zbl 0165.12601
[4] Gasymov, M. G.; Maksudov, F. G., The principal part of the resolvent of non-selfadjoint operators in neighbourhood of spectral singularities, Funct. Anal. Appl., 6, 185-192 (1972) · Zbl 0257.35067
[5] Greiner, W., Relativistic Quantum Mechanics, Wave Equations (1994), Springer-Verlag: Springer-Verlag Berlin/ New York
[6] Hruscev, S. V., Spectral singularities of dissipative Schrödinger operator with rapidly decreasing potential, Indiana Univ. Math. J., 33, 313-338 (1984) · Zbl 0548.34022
[7] Jaulent, M.; Jean, C., The inverse \(s\), Comm. Math. Phys., 28, 177-220 (1972)
[8] Jaulent, M.; Jean, C., The inverse problem for the one-dimensional Schrödinger equation with an energy-dependent potential I, II, Ann. Inst. H. Poincaré Sec. A, 25, 105-118 (1976) · Zbl 0357.34018
[9] Keldysh, M. V., On the completeness of the eigenfunctions of some classes of non-selfadjoint linear operators, Soviet Math. Dokl., 77, 11-14 (1951)
[10] Kemp, R. R.D., A singular boundary value problem for a non-selfadjoint differential operator, Canad. J. Math., 10, 447-462 (1958) · Zbl 0082.07402
[11] Krall, A. M., The adjoint of differential operator with integral boundary conditions, Proc. Amer. Math. Soc., 16, 738-742 (1965) · Zbl 0141.32302
[12] Krall, A. M., A nonhomogeneous eigenfunction expansion, Trans. Amer. Math. Soc., 117, 352-361 (1965) · Zbl 0141.32301
[13] Krall, A. M., Second order ordinary differential operators with general boundary conditions, Duke J. Math., 32, 617-625 (1965) · Zbl 0142.06201
[14] Krall, A. M., Nonhomogeneous differential operators, Michigan Math. J., 12, 247-265 (1965) · Zbl 0142.06202
[15] Krall, A. M., On non-selfadjoint ordinary differential operators of second order, Soviet Math. Dokl., 165, 1235-1237 (1965)
[16] Krall, A. M.; Bairamov, E.; Çakar, O., Spectrum and spectral singularities of quadratic pencil of Schrödinger operators with general boundary condition, J. Differential Equations, 151, 252-267 (1999) · Zbl 0927.34063
[17] Lax, P. D.; Philips, R. S., Scattering Theory (1967), Academic Press: Academic Press San Diego · Zbl 0214.12002
[18] Lyance, V. E., A Differential Operator with Spectral Singularities, I, II. A Differential Operator with Spectral Singularities, I, II, Amer. Math. Soc. Transl., Ser. 2, 60 (1967), Amer. Math. Soc: Amer. Math. Soc Providence
[19] Maksudov, F. G., Multiple expansion in the eigen and associoted functions of a quadratic pencil of one-dimensional singular differential operators, Spectral Theory of Operators (1977), p. 125-162
[20] Maksudov, F. G.; Allakhverdiev, B. P., Spectral analysis of a new class of non-selfadjoint operators with continuos and point spectrum, Soviet Math. Dokl., 30, 566-569 (1984) · Zbl 0598.47013
[21] Maksudov, F. G.; Guseinov, G. Sh., On solution of the inverse scattering problem for a quadratic pencil of one-dimensional Schrödinger operators on the whole axis, Soviet Math. Dokl., 34, 34-38 (1987) · Zbl 0629.34013
[22] Marchenko, V. A., Expansion in Eigenfunctions of Non-Selfadjoint Singular Second Order Differential Operators. Expansion in Eigenfunctions of Non-Selfadjoint Singular Second Order Differential Operators, Amer. Math. Soc. Transl. Ser. 2, 25 (1963), Amer. Math. Soc: Amer. Math. Soc Providence, p. 77-130 · Zbl 0127.03802
[23] Nagy, B. Sz.; Foias, C., Harmonic Analysis of Operators in Hilbert Space (1970), North-Holland: North-Holland Amsterdam · Zbl 0201.45003
[24] Naimark, M. A., Investigation of the spectrum and the expansion in eigenfunctions of a non-selfadjoint operator of second order on a semi-axis, Amer. Math. Soc. Transl. Ser. 2, 16, 103-193 (1960) · Zbl 0100.29504
[25] Naimark, M. A., Linear Differential Operators I, II (1968), Ungar: Ungar New York · Zbl 0227.34020
[26] Pavlov, B. S., The non-selfadjoint Schrödinger operator, Topics in Math. Phys., 1, 87-110 (1967)
[27] Pavlov, B. S., On separation conditions for the spectral components of a dissipative operators, Math. USSR Izvestiya, 9, 113-137 (1975) · Zbl 0323.47011
[28] Schwartz, J. T., Some non-selfadjoint operators, Comm. Pure Appl. Math., 13, 609-639 (1960) · Zbl 0096.08901
[29] Titchmarsh, E. C., Eigenfunction Expansions Associated with Second Order Differential Equations (1962), Oxford Univ. Press: Oxford Univ. Press London · Zbl 0099.05201
[30] Veliev, O. A., Spectral expansion of non-selfadjoint differential operators with periodic coefficients, J. Differential Equations, 22, 1403-1408 (1987) · Zbl 0665.34025
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