Abstract
We prove a theorem on the completeness of the system of root functions of the Schrödinger operator L = −d 2/dx 2 + p(x) on the half-line R+ with a potential p for which L appears to be maximal sectorial. An application of this theorem to the complex Airy operator L c = −d 2/dx 2 + cx, c = const, implies the completeness of the system of eigenfunctions of L c for the case in which |arg c| < 2π/3.We use subtler methods to prove a theorem stating that the system of eigenfunctions of this special operator remains complete under the condition that |arg c| < 5π/6.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. V. Keldysh, “On the eigenfunctions and eigenvalues for some classes of nonself-adjoint equations,” Dokl. Akad. Nauk SSSR, 77:1) (1951), 11–14.
A. A. Shkalikov, “Perturbations of self-adjoint and normal operators with discrete spectrum,” Uspekhi Mat. Nauk, 71:5) (2016), 113–174; English transl.: Russian Math. Surveys, 71:5 (2016), 907–964.
V. B. Lidskii, “A nonself-adjoint operator of Sturm–Liouville type with a discrete spectrum,” Trudy Moskov. Mat. Obshch., 9 (1960), 45–79.
E. B. Davies, “Wild spectral behaviour of anharmonic oscillators,” Bull. Lond. Math. Soc., 32:4) (2000), 432–438.
Materials of the workshop ”Mathematical aspects with non-self-adjoint operators”. A list of open problems, http://aimath.org/pastworkshops/nonselfadjoint-problems.pdf.
Y. Almog, “The stability of the normal state of superconductors in the presence of electric currents,” SIAM J. Math. Anal., 40:2) (2008), 824–850.
A. A. Shkalikov, “The limit behaviour of the spectrum for large parametric value in a model problem,” Mat. Zametki, 62:6) (1997), 950–953; English transl.: Math. Notes, 62:6 (1997), 796–799.
A. V. Dyachenko and A. A. Shkalikov, “On a model problem for the Orr–Zommerfeld equation with linear profile,” Funkts. Anal. Prilozhen., 36:3) (2002), 71–75; English transl.: Functional Anal. Appl., 36:3 (2002), 228–232.
S. N. Tumanov and A. A. Shkalikov, “On the spectrum localization of the Orr–Zommerfeld problem for large Reynolds Number,” Mat. Zametki, 72:4) (2002), 561–569; English transl.: Math. Notes, 72:4 (2002), 519–526.
A. A. Shkalikov, “Spectral portraits of the Orr–Sommerfeld operator,” J. Math. Sci., 124:6) (2004), 5417–5441.
L. N. Trefethen, M. Embree, Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators, Princeton Univ. Press, Princeton, 2005.
D. Krejčiřík, P. Siegl, M. Tater, J. Viola, “Pseudospectra in non–Hermitian quantum mechanics,” J. Math. Phys., 56 (2015), 503–513.
R. Henry, D. Krejčiřík, “Pseudospectra of the Schroedinger operator with a discontinuous complex potential,” J. Spectr. Theory (в печати); https://arxiv.org/abs/1503.02478.
J. Adduci, B. S. Mityagin, “Eigensystem of an L2-perturbed harmonic oscillator is an unconditional basis,” Cent. Eur. J. Math., 10:2) (2012), 569–589.
A. A. Shkalikov, “On the basis porperty of root vectors of a perturbed self-adjoint operator,” Trudy Mat. Inst. Steklov., 269 (2010), 290–303; English transl.: Proc. Steklov Inst. Math., 269 (2010), 284–298.
P. Djakov and B. S. Mityagin, “Riesz bases consisting of root functions of 1D Dirac operators,” Proc. Amer. Math. Soc., 141:4) (2013), 1361–1375.
B. S. Mityagin, “The spectrum of a harmonic oscillator operator perturbed by point interactions,” Int. J. Theor. Phys., 54:11) (2015), 4068–4085.
B. S. Mityagin, P. Siegl, and J. Viola, “Differential operators admitting various rates of spectral projection growth,” J. Func. Anal., 2017 (to appear); https://arxiv.org/abs/1309.3751.
B. S. Mityagin and P. Siegl, “Root system of singular perturbations of the harmonic oscillator type operators,” Lett. Math. Phys., 106:2) (2016), 147–167.
B. S. Mityagin and P. Siegl, Local form–subordination condition and Riesz basisness of root systems, https://arxiv.org/abs/1608.00224v1.
E. B. Davies, “Semi-classical states for non-self-adjoint Schrödinger operators,” Comm. Math. Phys., 200:1) (1999), 35–41.
S. N. Tumanov and A. A. Shkalikov, “On the limit behaviour of the spectrum of a model problem for the Orr–Sommerfeld equation with Poiseuille profile,” Izv. Ross. Akad. Nauk Ser. Mat., 66:4) (2002), 177–204; English transl.: Russian Acad. Sci. Izv. Math., 66:4 (2002), 829–856.
E. B. Davies and A. B. J. Kuijlaars, “Spectral asymptotics of the non-self-adjoint harmonic oscillator,” J. London Math. Soc., 70:2) (2004), 420–426.
R. Henry, “Spectral instability of some non-selfadjoint anharmonic oscillators,” C. R. Math. Acad. Sci. Paris, 350:23–24 (2012), 1043–1046.
R. Henry, “Spectral instability for even non-selfadjoint anharmonic oscillators,” J. Spectr. Theory, 4:2) (2014), 349–364.
R. Henry, “Spectral projections of the complex cubic oscillator,” Ann. Henri Poincaré, 15:10) (2014), 2025–2043.
C. M. Bender and S. Boettcher, “Real spectra in non-Hermitean Hamiltonians having PT symmetry,” Phys. Rev. Lett., 80:24 (1998), 5243.
P. Siegl and D. Krejčiřík, “On the metric operator for the imaginary cubic oscillator,” Phys. Rev., 86:12) (2012), 121702.
A. Eremenko, A. Gabrielov, and B. Shapiro, “High energy eigenfunctions of one-dimensional Schrödinger operators with polynomial potentials,” Comput. Methods Funct. Theory, 8:2) (2008), 513–529.
D. S. Grebenkov, B. Helffer, R. Henry, The complex Airy operator with a semi-permeable barries, https://arxiv.org/abs/1603.06992v1.
M. V. Fedoryuk, Asymptotic Analysis, Linear Ordinary Differential Equations, Springer–Verlag, Berlin, 1993.
M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, National Breau of Standards. Appl. Math. Series, vol. 52, Washington, 1972.
T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1980.
M. A. Naimark, Linear Differential Operators, Parts I, II, Ungar, New York, 1967, 1968.
B. Ya. Levin, Distribution of Zeros of Entire Functions, Amer. Math. Soc., Providence, RI, 1980.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 51, No. 1, pp. 82–98, 2017
Original Russian Text Copyright © by A. M. Savchuk and A. A. Shkalikov
Supported by RFBR grant no. 16-01-00706.
Rights and permissions
About this article
Cite this article
Savchuk, A.M., Shkalikov, A.A. Spectral properties of the complex airy operator on the half-line. Funct Anal Its Appl 51, 66–79 (2017). https://doi.org/10.1007/s10688-017-0168-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10688-017-0168-1