Hill’s operators with the potentials analytically dependent on energy. (English) Zbl 1520.47084
Summary: We consider Schrödinger operators on the line with potentials that are periodic with respect to the coordinate variable and real analytic with respect to the energy variable. We prove that if the imaginary part of the potential is bounded in the right half-plane, then the high energy spectrum is real, and the corresponding asymptotics are determined. Moreover, the Dirichlet and Neumann problems are considered. These results are used to analyze the good Boussinesq equation.
MSC:
| 47E05 | General theory of ordinary differential operators |
| 34L20 | Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators |
| 34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |
References:
| [1] | Alonso, L. M., Schrödinger spectral problems with energy-dependent potentials as sources of nonlinear Hamiltonian evolution equations, J. Math. Phys., 21, 9, 2342-2349 (1980) · Zbl 0455.35111 |
| [2] | Badanin, A.; Korotyaev, E., Spectral estimates for periodic fourth order operators, St. Petersburg Math. J., 22, 5, 703-736 (2011) · Zbl 1230.34071 |
| [3] | Badanin, A.; Korotyaev, E. L., Even order periodic operators on the real line, Int. Math. Res. Not., 2012, 5, 1143-1194 (2012) · Zbl 1239.34101 |
| [4] | Badanin, A.; Korotyaev, E., Third order operator with periodic coefficients on the real axis, St. Petersburg Math. J., 25, 5, 713-734 (2014) · Zbl 1310.34116 |
| [5] | Badanin, A.; Korotyaev, E., Spectral asymptotics for the third order operator with periodic coefficients, J. Differ. Equ., 253, 3113-3146 (2012) · Zbl 1273.47074 |
| [6] | Badanin, A.; Korotyaev, E., Third-order operators with three-point conditions associated with Boussinesq’s equation, Appl. Anal. (2019) |
| [7] | Calogero, F.; Jagannathan, G., Levinson’s theorem for energy-dependent potentials, Nuovo Cimento A, 47, 2, 178-188 (1967) |
| [8] | Derkach, V. A.; Malamud, M. M., Some classes of analytic operator-valued functions with a nonnegative imaginary part, Dokl. Akad. Nauk Ukr. SSR, Ser. A., 3, 13-17 (1989), 87 · Zbl 0692.47016 |
| [9] | Formánek, J.; Lombard, R. J.; Mareš, J., Wave equations with energy-dependent potentials, Czechoslov. J. Phys., 54, 3, 289-315 (2004) |
| [10] | Gesztesy, F.; Kalton, N. J.; Makarov, K. A.; Tsekanovskii, E., Some applications of operator-valued Herglotz functions, (Operator Theory, System Theory and Related Topics (2001), Birkhäuser: Birkhäuser Basel), 271-321 · Zbl 0991.30020 |
| [11] | Jaulent, M.; Jean, C., The inverse problem for the one-dimensional Schrödinger equation with an energy-dependent potential, I, Ann. Inst. Henri Poincaré, 25, 2, 105-118 (1976) · Zbl 0357.34018 |
| [12] | Jaulent, M.; Jean, C., The inverse problem for the one-dimensional Schrödinger equation with an energy-dependent potential. II, Ann. Inst. Henri Poincaré, 25, 2, 119-137 (1976) · Zbl 0357.34019 |
| [13] | Kamimura, Y., Energy dependent inverse scattering on the line, Differ. Integral Equ., 21, 11-12, 1083-1112 (2008) · Zbl 1224.81017 |
| [14] | Keldysh, M. V., On the completeness of the eigenfunctions of some classes of non-selfadjoint linear operators, Russ. Math. Surv., 26, 4, 15-44 (1971) · Zbl 0225.47008 |
| [15] | Korotyaev, E., Inverse problem and the trace formula for the Hill operator, II, Math. Z., 231, 2, 345-368 (1999) · Zbl 0929.34016 |
| [16] | Korotyaev, E., Characterization of the spectrum of Schrödinger operators with periodic distributions, Int. Math. Res. Not., 37, 2019-2031 (2003) · Zbl 1104.34059 |
| [17] | Krein, M. G., The basic propositions of the theory of λ-zones of stability of a canonical system of linear differential equations with periodic coefficients, (Topics in Differential and Integral Equations and Operator Theory (1983), Birkhäuser: Birkhäuser Basel), 1-105 · Zbl 0512.45001 |
| [18] | Markus, A. S., Introduction to the Spectral Theory of Polynomial Operator Pencils (2012), American Mathematical Soc. |
| [19] | McKean, H., Boussinesq’s equation on the circle, Commun. Pure Appl. Math., 34, 599-691 (1981) · Zbl 0473.35070 |
| [20] | Papanicolaou, V., The spectral theory of the vibrating periodic beam, Commun. Math. Phys., 170, 359-373 (1995) · Zbl 0828.34078 |
| [21] | Papanicolaou, V., The periodic Euler-Bernoulli equation, Trans. Am. Math. Soc., 355, 9, 3727-3759 (2003) · Zbl 1052.34079 |
| [22] | Pöschel, J.; Trubowitz, E., Inverse Spectral Theory (1987), Academic Press: Academic Press Boston · Zbl 0623.34001 |
| [23] | Reed, M.; Simon, B., Methods of Modern Mathematical Physics. IV. Analysis of Operators (1978), Academic Press: Academic Press New York-London · Zbl 0401.47001 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
