Singular Schrödinger operators as self-adjoint extensions of 𝑁-entire operators

Silva, Luis; Teschl, Gerald; Toloza, Julio

doi:10.1090/S0002-9939-2014-12440-3

Singular Schrödinger operators as self-adjoint extensions of $N$-entire operators
HTML articles powered by AMS MathViewer

by Luis O. Silva, Gerald Teschl and Julio H. Toloza PDF

Proc. Amer. Math. Soc. 143 (2015), 2103-2115 Request permission

Abstract:

We investigate the connections between Weyl–Titchmarsh– Kodaira theory for one-dimensional Schrödinger operators and the theory of $n$-entire operators. As our main result we find a necessary and sufficient condition for a one-dimensional Schrödinger operator to be $n$-entire in terms of square integrability of derivatives (w.r.t. the spectral parameter) of the Weyl solution. We also show that this is equivalent to the Weyl function being in a generalized Herglotz–Nevanlinna class. As an application we show that perturbed Bessel operators are $n$-entire, improving the previously known conditions on the perturbation.

References

S. Albeverio, R. Hryniv, and Ya. Mykytyuk, Inverse spectral problems for Bessel operators, J. Differential Equations 241 (2007), no. 1, 130–159. MR 2356213, DOI 10.1016/j.jde.2007.06.017
Chris Brislawn, Kernels of trace class operators, Proc. Amer. Math. Soc. 104 (1988), no. 4, 1181–1190. MR 929421, DOI 10.1090/S0002-9939-1988-0929421-X
W. Bulla and F. Gesztesy, Deficiency indices and singular boundary conditions in quantum mechanics, J. Math. Phys. 26 (1985), no. 10, 2520–2528. MR 803795, DOI 10.1063/1.526768
Robert Carlson, Inverse spectral theory for some singular Sturm-Liouville problems, J. Differential Equations 106 (1993), no. 1, 121–140. MR 1249180, DOI 10.1006/jdeq.1993.1102
Robert Carlson, A Borg-Levinson theorem for Bessel operators, Pacific J. Math. 177 (1997), no. 1, 1–26. MR 1444769, DOI 10.2140/pjm.1997.177.1
Louis de Branges, Hilbert spaces of entire functions, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1968. MR 0229011
Jonathan Eckhardt, Inverse uniqueness results for Schrödinger operators using de Branges theory, Complex Anal. Oper. Theory 8 (2014), no. 1, 37–50. MR 3147711, DOI 10.1007/s11785-012-0265-3
Jonathan Eckhardt and Gerald Teschl, Uniqueness results for one-dimensional Schrödinger operators with purely discrete spectra, Trans. Amer. Math. Soc. 365 (2013), no. 7, 3923–3942. MR 3042609, DOI 10.1090/S0002-9947-2012-05821-1
W. N. Everitt and H. Kalf, The Bessel differential equation and the Hankel transform, J. Comput. Appl. Math. 208 (2007), no. 1, 3–19. MR 2347733, DOI 10.1016/j.cam.2006.10.029
Fritz Gesztesy and Maxim Zinchenko, On spectral theory for Schrödinger operators with strongly singular potentials, Math. Nachr. 279 (2006), no. 9-10, 1041–1082. MR 2242965, DOI 10.1002/mana.200510410
J.-C. Guillot and J. V. Ralston, Inverse spectral theory for a singular Sturm-Liouville operator on $[0,1]$, J. Differential Equations 76 (1988), no. 2, 353–373. MR 969430, DOI 10.1016/0022-0396(88)90080-0
Aleksey Kostenko, Alexander Sakhnovich, and Gerald Teschl, Inverse eigenvalue problems for perturbed spherical Schrödinger operators, Inverse Problems 26 (2010), no. 10, 105013, 14. MR 2719774, DOI 10.1088/0266-5611/26/10/105013
Aleksey Kostenko, Alexander Sakhnovich, and Gerald Teschl, Weyl-Titchmarsh theory for Schrödinger operators with strongly singular potentials, Int. Math. Res. Not. IMRN 8 (2012), 1699–1747. MR 2920828, DOI 10.1093/imrn/rnr065
Aleksey Kostenko, Alexander Sakhnovich, and Gerald Teschl, Commutation methods for Schrödinger operators with strongly singular potentials, Math. Nachr. 285 (2012), no. 4, 392–410. MR 2899633, DOI 10.1002/mana.201000108
Aleksey Kostenko and Gerald Teschl, On the singular Weyl-Titchmarsh function of perturbed spherical Schrödinger operators, J. Differential Equations 250 (2011), no. 9, 3701–3739. MR 2773183, DOI 10.1016/j.jde.2010.10.026
Matthias Langer and Harald Woracek, A characterization of intermediate Weyl coefficients, Monatsh. Math. 135 (2002), no. 2, 137–155. MR 1894093, DOI 10.1007/s006050200011
Matthias Lesch and Boris Vertman, Regular singular Sturm-Liouville operators and their zeta-determinants, J. Funct. Anal. 261 (2011), no. 2, 408–450. MR 2793118, DOI 10.1016/j.jfa.2011.03.011
Michael Reed and Barry Simon, Methods of modern mathematical physics. III, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. Scattering theory. MR 529429
Frédéric Serier, The inverse spectral problem for radial Schrödinger operators on $[0,1]$, J. Differential Equations 235 (2007), no. 1, 101–126. MR 2309569, DOI 10.1016/j.jde.2006.12.014
Luis O. Silva and Julio H. Toloza, The class of $n$-entire operators, J. Phys. A 46 (2013), no. 2, 025202, 23. MR 3002855, DOI 10.1088/1751-8113/46/2/025202
Luis O. Silva and Julio H. Toloza, A class of $n$-entire Schrödinger operators, Complex Anal. Oper. Theory 8 (2014), no. 8, 1581–1599. MR 3275436, DOI 10.1007/s11785-013-0329-z
Gerald Teschl, Mathematical methods in quantum mechanics, Graduate Studies in Mathematics, vol. 99, American Mathematical Society, Providence, RI, 2009. With applications to Schrödinger operators. MR 2499016, DOI 10.1090/gsm/099
Joachim Weidmann, Spectral theory of ordinary differential operators, Lecture Notes in Mathematics, vol. 1258, Springer-Verlag, Berlin, 1987. MR 923320, DOI 10.1007/BFb0077960