zbMATH — the first resource for mathematics

Spectral asymptotics for perturbed spherical Schrödinger operators and applications to quantum scattering. (English) Zbl 1284.34130
The authors investigate perturbed spherical Schrödinger operators with a real-valued potential in an arbitrary space dimension. They first establish the high energy asymptotics of the singular Weyl function and the associated spectral measure for those operators. Their innovative approach uses a one-term asymptotic formula for the \(m\)-function, along with the commutation method for Bessel operators.
A Liouville type transform is used in the proof. The authors then enumerate examples of the usefulness of their findings including, among them the application to the quantum mechanical scattering problem. Results on the asymptotic behavior of \(m\)-functions of regular Sturm-Liouville problems are listed in the appendix.

34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34L05 General spectral theory of ordinary differential operators
34B20 Weyl theory and its generalizations for ordinary differential equations
Full Text: DOI arXiv
[1] Albeverio, S.; Hryniv, R.; Mykytyuk, Ya., Inverse spectral problems for Bessel operators, J. Diff. Eqs., 241, 130-159, (2007) · Zbl 1129.34004
[2] Albeverio, S.; Hryniv, R.; Mykytyuk, Ya., Scattering theory for Schrödinger operators with Bessel-type potentials, J. Reine Angew. Math., 666, 83-113, (2012) · Zbl 1253.47007
[3] Bennewitz, C., Spectral asymptotics for Sturm-Liouville equations, Proc. London Math. Soc. (3), 59, 294-338, (1989) · Zbl 0681.34023
[4] Chadan, K., Sabatier, P.C.: Inverse Problems in Quantum Scattering Theory. Berlin-Heidelberg, Newyork: Springer-Verlag, 1989 · Zbl 0681.35088
[5] Cheeger, J.; Taylor, M., On the diffraction of waves by conical singularities, I, Comm. Pure Appl. Math., 25, 275-331, (1982) · Zbl 0526.58049
[6] Coz, M.; Coudray, C., The Riemann solution and the inverse quantum mechanical problem, J. Math. Phys., 17, 888-893, (1976)
[7] Derkach, V., Extensions of Laguerre operators in indefinite inner product spaces, Math. Notes, 63, 449-459, (1998) · Zbl 0923.47018
[8] Dijksma, A.; Shondin, Yu., Singular point-like perturbations of the Bessel operator in a Pontryagin space, J. Diff. Eqs., 164, 49-91, (2000) · Zbl 0960.47021
[9] Djurčić, D.; Togašev, A.; Ješić, J., The strong asymptotic equivalence and the generalized inverse, Sib. Math. J., 49, 628-636, (2008)
[10] Eckhardt, J.: Inverse uniqueness results for Schrödinger operators using de Branges theory, Complex Anal. Oper. Theory, to appear, doi:10.1007s11785-012-0265-3, 2012
[11] Eckhardt, J., Teschl, G.: Uniqueness results for one-dimensional Schrödinger operators with purely discrete spectra. Trans. Amer. Math. Soc. (to appear), doi:10.1090/S0002-9947-2012-05821-1, 2012 · Zbl 1230.34077
[12] Everitt, W.N., On a property of the \(m\)-coefficient of a second order linear differential equation, J. London Math. Soc. (2), 4, 443-457, (1972) · Zbl 0262.34012
[13] Faddeev, L., The inverse problem in quantum scattering theory, J. Math. Phys., 4, 72-104, (1963) · Zbl 0112.45101
[14] Fulton, C., Titchmarsh-Weyl \(m\)-functions for second order Sturm-Liouville problems, Math. Nachr., 281, 1417-1475, (2008) · Zbl 1165.34011
[15] Fulton, C.; Langer, H., Sturm-Liouville operators with singularities and generalized Nevanlinna functions, Complex Anal. Oper. Theory, 4, 179-243, (2010) · Zbl 1214.34022
[16] Fulton, C.; Langer, H.; Luger, A., Mark krein’s method of directing functionals and singular potentials, Math. Nachr., 1285, 1791-1798, (2012) · Zbl 1259.34017
[17] Gesztesy, F.; Plessas, W.; Thaller, B., On the high-energy behaviour of scattering phase shifts for Coulomb-like potentials, J. Phys. A: Math. Gen., 13, 2659-2671, (1980)
[18] Gesztesy, F.; Thaller, B., Born expansions for Coulomb-type interactions, J. Phys. A: Math. Gen., 14, 639-657, (1981)
[19] Gesztesy, F.; Zinchenko, M., On spectral theory for Schrödinger operators with strongly singular potentials, Math. Nachr., 279, 1041-1082, (2006) · Zbl 1108.34063
[20] Guillot, J.-C.; Ralston, J.V., Inverse spectral theory for a singular Sturm-Liouville operator on [0,1], J. Diff. Eqs., 76, 353-373, (1988) · Zbl 0688.34013
[21] Hille, E., Green’s transform and singular boundary value problems, J. Math. Pures Appl. (9), 42, 331-349, (1963) · Zbl 0121.31101
[22] Jost, R., Über die falschen nullstellen der eigenwerte der \(S\)-matrix, Helv. Phys. Acta, 20, 256-266, (1947)
[23] Kac, I.S.: The existence of spectral functions of generalized second order differential systems with boundary conditions at the singular end. Mat. Sbornik 68, 174-227 (1965); English transl. in: Amer. Math. Soc. Transl. Ser. (2) 62, 204-262 (1967) · Zbl 0112.45101
[24] Kac, I.S.: Integral characteristics of the growth of spectral functions for generalized second order boundary problems with boundary conditions at a regular end. Izv. Akad. Nauk SSSR, Ser. Mat. 35, 154-184 (1971); English transl. in: Math. Ussr-Izv. 5, 161-191 (1973) · Zbl 0090.19303
[25] Kac, I.S.: A generalization of the asymptotic formula of V.A. Marchenko for the spectral function of a second order boundary value problem. Izv. Akad. Nauk SSSR, Ser. Mat. 37, 422-436 (1973); English transl. in: Math. Ussr-Izv. 7, 424-436 (1973)
[26] Kac, I.S.; Krein, M.G., On the spectral function of the string, Amer. Math. Soc. Transl. Ser. (2), 103, 19-102, (1974) · Zbl 0291.34017
[27] Kasahara, Y., Spectral theory of generalized second order differential operators and its applications to Markov processes, Japan J. Math., 1, 67-84, (1975) · Zbl 0348.60113
[28] Kodaira, K., The eigenvalue problem for ordinary differential equations of the second order and heisenberg’s theory of \(S\)-matrices, Amer. J. Math., 71, 921-945, (1949) · Zbl 0035.27101
[29] Koosis, P.: Introduction to H_{\(p\)}Spaces. 2\^{}{nd} ed., Cambridge: Cambridge UP, 1998 · Zbl 1024.30001
[30] Kostenko, A., Sakhnovich, A., Teschl, G.: Inverse eigenvalue problems for perturbed spherical Schrödinger operators. Inverse Problems 26, 105013, 14pp (2010) · Zbl 1227.34023
[31] Kostenko, A.; Sakhnovich, A.; Teschl, G., Weyl-titchmarsh theory for Schrödinger operators with strongly singular potentials, Int. Math. Res. Not., 2012, 1699-1747, (2012) · Zbl 1248.34027
[32] Kostenko, A.; Sakhnovich, A.; Teschl, G., Commutation methods for Schrödinger operators with strongly singular potentials, Math. Nachr., 285, 392-410, (2012) · Zbl 1250.34069
[33] Kostenko, A.; Teschl, G., On the singular Weyl-titchmarsh function of perturbed spherical Schrödinger operators, J. Diff. Eqs., 250, 3701-3739, (2011) · Zbl 1219.34037
[34] Kurasov, P.; Luger, A., An operator theoretic interpretation of the generalized titchmarsh-Weyl coefficient for a singular Sturm-Liouville problem, Math. Phys. Anal. Geom., 14, 115-151, (2011) · Zbl 1267.34052
[35] Lesch, M.; Vertman, B., Regular-singular Sturm-Liouville operators and their zeta-determinants, J. Funct. Anal., 261, 408-450, (2011) · Zbl 1230.34077
[36] Marchenko, V.A.: Some questions in the theory of one-dimensional linear differential operators of the second order, I. Trudy Moskov. Mat. Obsc. 1, 372-420 (1952); English. transl. in: Amer. Math. Soc. Transl. Series (2), 101, 1-104 (1972)
[37] Marchenko, V.A.: Sturm-Liouville Operators and Applications: Revised Edition. Providence, RI: Amer. Math. Soc./Chelsea Publ., 2011 · Zbl 1298.34001
[38] Newton, R.G., Analytic properties of radial wave functions, J. Math. Phys., 1, 319-347, (1960) · Zbl 0090.19303
[39] Olver et al., F.W.J.: NIST Handbook of Mathematical Functions. Cambridge: Cambridge University Press, 2010 · Zbl 1198.00002
[40] Serier, F., The inverse spectral problem for radial Schrödinger operators on [0, 1], J. Diff. Eqs., 235, 101-126, (2007) · Zbl 1119.34006
[41] Teschl, G.: Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators. Graduate Studies in Mathematics, Providence, RI: Amer. Math. Soc., 2009 · Zbl 1166.81004
[42] Weidmann, J.: Spectral Theory of Ordinary Differential Operators. Lecture Notes in Mathematics 1258. Berlin: Springer, 1987 · Zbl 0647.47052
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.