Abstract
We derive an asymptotic expansion for the Weyl function of a one-dimensional Schrödinger operator which generalizes the classical formula by Atkinson. Moreover, we show that the asymptotic formula can also be interpreted in the sense of distributions.
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Albeverio, S., Gesztesy, F., Høegh-Krohn, R., Holden, H.: Solvable Models in Quantum Mechanics, 2nd edn. AMS Chelsea Publishing, Providence (2005)
Atkinson, F.V.: On the location of Weyl circles. Proc. R. Soc. Edinb. 88A, 345–356 (1981)
Ben Amor, A., Remling, C.: Direct and inverse spectral theory of one-dimensional Schrödinger operators with measures. Integr. Equ. Oper. Theory 52(3), 395–417 (2005)
Bennewitz, C.: A note on the Titchmarsh–Weyl \(m\)-function, vol. 2, pp. 105–111. Argonne Nat. Lab (preprint, ANL-87-26) (1988)
Bennewitz, C.: Spectral asymptotics for Sturm–Liouville equations. Proc. Lond. Math. Soc. (3) 59(2), 294–338 (1989)
Boutet de Monvel, A., Marchenko, V.: Asymptotic formulas for spectral and Weyl functions of Sturm–Liouville operators with smooth potentials. In: Gohberg, I., Lubich, Y. (eds.) New Results in Operator Theory and its Applications, Operator Theory, Advances and Applications, vol. 98, pp. 102–117. Birkhäuser, Basel (1997)
Danielyan, A.A., Levitan, B.M.: On the asymptotic behavior of the Weyl–Titchmarsh \(m\)-function. Math. USSR Izv. 36, 487–496 (1991)
Eckhardt, J., Gesztesy, F., Nichols, R., Teschl, G.: Supersymmetry and Schrödinger-type operators with distributional matrix-valued potentials. J. Spectr. Theory 4, 715–768 (2014)
Eckhardt, J., Gesztesy, F., Nichols, R., Teschl, G.: Weyl–Titchmarsh theory for Sturm–Liouville operators with distributional potentials. Opusc. Math. 33, 467–563 (2013)
Eckhardt, J., Gesztesy, F., Nichols, R., Teschl, G.: Inverse spectral theory for Sturm-Liouville operators with distributional potentials. J. Lond. Math. Soc. 88(2), 801–828 (2013)
Eckhardt, J., Teschl, G.: Sturm–Liouville operators with measure-valued coefficients. J. Anal. Math. 120, 151–224 (2013)
Everitt, W.N.: On a property of the \(m\)-coefficient of a second-order linear differential equation. J. Lond. Math. Soc. 2(4), 443–457 (1972)
Everitt, W.N., Halvorsen, S.G.: On the asymptotic form of the Titchmarsh–Weyl \(m\)-coefficient. Appl. Anal. 8, 153–169 (1978)
Everitt, W.N., Hinton, D.B., Shaw, J.K.: The asymptotic form of the Titchmarsh–Weyl coefficient for Dirac systems. J. Lond. Math. Soc. 2(27), 465–476 (1983)
Harris, B.J.: On the Titchmarsh–Weyl \(m\)-function. Proc. R. Soc. Edinb. 95A, 223–237 (1983)
Harris, B.J.: The asymptotic form of the Titchmarsh–Weyl \(m\)-function. J. Lond. Math. Soc. 2(30), 110–118 (1984)
Harris, B.J.: The asymptotic form of the Titchmarsh–Weyl \(m\)-function associated with a second order differential equation with locally integrable coefficient. Proc. R. Soc. Edinb. 102A, 243–251 (1986)
Harris, B.J.: An exact method for the calculation of certain Titchmarsh–Weyl \(m\)-functions. Proc. R. Soc. Edinb. 106A, 137–142 (1987)
Hille, E.: Lectures on Ordinary Differential Equations. Addison-Wesley, Reading (1969)
Hinton, D.B., Klaus, M., Shaw, J.K.: Series representation and asymptotics for Titchmarsh–Weyl \(m\)-functions. Differ. Integral Equ. 2, 419–429 (1989)
Kaper, H.G., Kwong, M.M.: Asymptotics of the Titchmarsh–Weyl \(m\)-coefficient for integrable potentials. Proc. R. Soc. Edinb. 103A, 347–358 (1986)
Kaper, H.G., Kwong, M.M.: Asymptotics of the Titchmarsh–Weyl \(m\)-coefficient for integrable potentials, II. In: Knowles, W., Saito, Y. (eds.) Differential Equations and Mathematical Physics, I. Lecture Notes in Mathematics, vol. 1285, pp. 222–229. Springer, Berlin (1987)
Levitan, B.M.: A remark on one theorem of V. A. Marchenko. Trudy Moskov. Mat. Obsch. 1, 421–422 (1952). (In Russian)
Levitan, B.M., Sargsjan, I.S.: Sturm–Liouville and Dirac Operators (Russian). Nauka, Moscow (1988)
Marchenko, V.A.: Some questions in the theory of one-dimensional second-order linear differential operators. I. Trudy Moskov. Mat. Obsch. 1, 327–420 (1952) (In Russian). Am. Math. Soc. Transl. (2) 101, 1–104 (1973)
Pechentsov, A.S.: Trace of a difference of singular Sturm–Liouville operators with a potential containing Dirac \(\delta \)-functions. Russ. J. Math. Phys. 20, 230–238 (2013)
Rybkin, A.: On the trace approach to the inverse scattering problem in dimension one. SIAM J. Math. Anal. 32, 1248–1264 (2001)
Savchuk, A.M., Shkalikov, A.A.: Trace formula for Sturm–Liouville operators with singular potentials. Math. Notes 69, 387–400 (2001)
Simon, B.: A new approach to inverse spectral theory, I. Fundamental formalism. Ann. Math. 150, 1029–1057 (1999)
Teschl, G.: Mathematical methods in quantum mechanics; with applications to Schrödinger operators, 2nd edn, vol. 157. Graduate Studies in Mathematics, American Mathematical Society, RI (2014)
Weyl, H.: Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen. Math. Ann. 68, 220–269 (1910)
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We are indebted to Jonathan Eckhardt, Fritz Gesztesy and Helge Holden for discussions on this subject.
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Communicated by A. Constantin.
Research supported by the Austrian Science Fund (FWF) under Grant No. Y330.
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Luger, A., Teschl, G. & Wöhrer, T. Asymptotics of the Weyl function for Schrödinger operators with measure-valued potentials. Monatsh Math 179, 603–613 (2016). https://doi.org/10.1007/s00605-015-0740-9
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DOI: https://doi.org/10.1007/s00605-015-0740-9