Geodesic
Spectral Properties of the Complex Airy Operator on the Half-Line
Funkcionalʹnyj analiz i ego priloženiâ, Tome 51 (2017) no. 1, pp. 82-98.

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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 $\mathbb R_+$ with a potential $p$ for which $L$ appears to be maximal sectorial. An application of this theorem to the complex Airy operator $\mathcal L_c = - d^2\!/dx^2 +cx$, $c=\operatorname{const}$, implies the completeness of the system of eigenfunctions of $\mathcal L_c$ for the case in which $|\arg c| 2\pi/3$. We use subtler methods to prove a theorem stating that the system of eigenfunctions of this special operator remains complete under the condition that $|\arg c| 5\pi/6$.
Keywords: Schrödinger operator, complex Airy operator, nonself-adjoint operator, completeness of the eigenfunctions of a differential operator. 
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A. M. Savchuk; A. A. Shkalikov. Spectral Properties of the Complex Airy Operator on the Half-Line. Funkcionalʹnyj analiz i ego priloženiâ, Tome 51 (2017) no. 1, pp. 82-98. http://geodesic.mathdoc.fr/item/FAA_2017_51_1_a5/

[1] M. V. Keldysh, “O sobstvennykh znacheniyakh i sobstvennykh funktsiyakh nekotorykh klassov nesamosopryazhennykh uravnenii”, Dokl. AN SSSR, 77:1 (1951), 11–14 | Zbl

[2] A. A. Shkalikov, “Vozmuscheniya samosopryazhennykh i normalnykh operatorov s diskretnym spektrom”, UMN, 71:5 (2016), 113–174 | DOI | MR | Zbl

[3] V. B. Lidskii, “Nesamosopryazhennyi operator tipa Shturma–Liuvillya s diskretnym spektrom”, Trudy MMO, 9 (1960), 45–79 | Zbl

[4] E. B. Davies, “Wild spectral behaviour of anharmonic oscillators”, Bull. Lond. Math. Soc., 32:4 (2000), 432–438 | DOI | MR | Zbl

[5] A list of open problems,, Materials of the workshop “Mathematical aspects with non-self-adjoint operators” http://aimath.org/pastworkshops/nonselfadjoint-problems.pdf

[6] 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 | DOI | MR | Zbl

[7] A. A. Shkalikov, “O predelnom povedenii spektra pri bolshikh znacheniyakh parametra odnoi modelnoi zadachi”, Matem. zametki, 62:6 (1997), 950–953 | DOI | Zbl

[8] A. V. Dyachenko, A. A. Shkalikov, “O modelnoi zadache dlya uravneniya Orra–Zommerfelda s lineinym profilem”, Funkts. analiz i ego pril., 36:3 (2002), 71–75 | DOI | MR | Zbl

[9] S. N. Tumanov, A. A. Shkalikov, “O lokalizatsii spektra zadachi Orra–Zommerfelda dlya bolshikh chisel Reinoldsa”, Mat. zametki, 72:4 (2002), 561–569 | DOI | MR | Zbl

[10] A. A. Shkalikov, “Spectral portraits of the Orr–Sommerfeld operator”, J. Math. Sci., 124:6 (2004), 5417–5441 | DOI | MR

[11] L. N. Trefethen, M. Embree, Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators, Princeton Univ. Press, Princeton, 2005 | MR | Zbl

[12] D. Krejc̆ir̆ík, P. Siegl, M. Tater, J. Viola, “Pseudospectra in non–Hermitian quantum mechanics”, J. Math. Phys., 56 (2015), 503–513 | MR

[13] R. Henry, D. Krejc̆ir̆ík, “Pseudospectra of the Schroedinger operator with a discontinuous complex potential”, J. Spectr. Theory (to appear) | MR

[14] J. Adduci, B. S. Mityagin, “Eigensystem of an $L_2$-perturbed harmonic oscillator is an unconditional basis”, Cent. Eur. J. Math., 10:2 (2012), 569–589 | DOI | MR | Zbl

[15] A. A. Shkalikov, “O bazisnosti kornevykh vektorov vozmuschennogo samosopryazhennogo operatora”, Trudy MIAN, 269 (2010), 290–303 | Zbl

[16] P. Djakov, B. S. Mityagin, “Riesz bases consisting of root functions of 1D Dirac operators”, Proc. Amer. Math. Soc., 141:4 (2013), 1361–1375 | DOI | MR | Zbl

[17] B. S. Mityagin, “The spectrum of a harmonic oscillator operator perturbed by point interactions”, Int. J. Theor. Phys., 54:11 (2015), 4068–4085 | DOI | MR | Zbl

[18] B. S. Mityagin, P. Siegl, J. Viola, “Differential operators admitting various rates of spectral projection growth”, J. Func. Anal., 2017 (to appear) ; arXiv: 1309.3751 | MR

[19] B. S. Mityagin, P. Siegl, “Root system of singular perturbations of the harmonic oscillator type operators”, Lett. Math. Phys., 106:2 (2016), 147–167 | DOI | MR | Zbl

[20] B. S. Mityagin, P. Siegl, Local form–subordination condition and Riesz basisness of root systems, arXiv: 1608.00224v1

[21] E. B. Davies, “Semi-classical states for non-self-adjoint Schrödinger operators”, Comm. Math. Phys., 200:1 (1999), 35–41 | DOI | MR | Zbl

[22] S. N. Tumanov, A. A. Shkalikov, “O predelnom povedenii spektra modelnoi zadachi dlya uravneniya Orra–Zommerfelda s profilem Puazeilya”, Izv. RAN, ser. matem., 66:4 (2002), 177–204 | DOI | MR | Zbl

[23] E. B. Davies, A. B. J. Kuijlaars, “Spectral asymptotics of the non-self-adjoint harmonic oscillator”, J. London Math. Soc., 70:2 (2004), 420–426 | DOI | MR | Zbl

[24] R. Henry, “Spectral instability of some non-selfadjoint anharmonic oscillators”, C. R. Math. Acad. Sci. Paris, 350:23–24 (2012), 1043–1046 | DOI | MR | Zbl

[25] R. Henry, “Spectral instability for even non-selfadjoint anharmonic oscillators”, J. Spectr. Theory, 4:2 (2014), 349–364 | DOI | MR | Zbl

[26] R. Henry, “Spectral projections of the complex cubic oscillator”, Ann. Henri Poincaré, 15:10 (2014), 2025–2043 | DOI | MR | Zbl

[27] C. M. Bender, S. Boettcher, “Real spectra in non-Hermitean Hamiltonians having $\mathcal{P\!T}$ symmetry”, Phys. Rev. Lett., 80:24 (1998), 5243 | DOI | MR | Zbl

[28] P. Siegl, D. Krejc̆ir̆ík, “On the metric operator for the imaginary cubic oscillator”, Phys. Rev., 86:12 (2012), 121702

[29] A. Eremenko, A. Gabrielov, B. Shapiro, “High energy eigenfunctions of one-dimensional Schrödinger operators with polynomial potentials”, Comput. Methods Funct. Theory, 8:2 (2008), 513–529 | DOI | MR | Zbl

[30] D. S. Grebenkov, B. Helffer, R. Henry, The complex Airy operator with a semi-permeable barries,, arXiv: 1603.06992v1

[31] M. V. Fedoryuk, Asimptoticheskie metody dlya lineinykh obyknovennykh differentsialnykh uravnenii, Nauka, M., 1983 | MR

[32] M. Abramowitz, I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Appl. Math. Series, 52, National Breau of Standards, Washington, 1972 | MR

[33] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1980 | MR | Zbl

[34] M. A. Naimark, Lineinye differentsialnye operatory, Nauka, M., 1969

[35] B. Ya. Levin, Raspredelenie kornei tselykh funktsii, GITTL, M., 1956