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Pseudospectra in non-Hermitian quantum mechanics. (English) Zbl 1328.81116

Let \( H \) be the harmonic oscillator Hamiltonian perturbed by imaginary shift. The authors derive a lower estimate for the resolvent of \( H \) of the form \( \ln \| ( H - z )^{ -1 } \| \geq C \sqrt{ \operatorname{Re} z } \), \( C > 0 \), in a parabolic region of the right half plane by semiclassical methods.

MSC:

81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
47E05 General theory of ordinary differential operators
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)

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[3] Adams, R. A., Sobolev Spaces (1975) · Zbl 0314.46030
[4] Adduci, J.; Mityagin, B., Eigensystem of an \(L^2\)-perturbed harmonic oscillator is an unconditional basis, Cent. Eur. J. Math., 10, 569-589 (2012) · Zbl 1259.47059 · doi:10.2478/s11533-011-0139-3
[5] Adduci, J.; Mityagin, B., Root system of a perturbation of a selfadjoint operator with discrete spectrum, Integr. Equations Oper. Theory, 73, 153-175 (2012) · Zbl 1294.47022 · doi:10.1007/s00020-012-1967-7
[6] Ahmed, Z., Pseudo-Hermiticity of Hamiltonians under gauge-like transformation: Real spectrum of non-Hermitian Hamiltonians, Phys. Lett. A, 294, 287-291 (2002) · Zbl 0990.81029 · doi:10.1016/S0375-9601(02)00124-X
[7] Albeverio, S.; Guenther, U.; Kuzhel, S., J-self-adjoint operators with \(<mml:math display=''inline`` overflow=''scroll``> <mml:mi mathvariant=''script``>C\)-symmetries: Extension theory approach, J. Phys. A: Math. Theor., 42, 105205 (2009) · Zbl 1162.47060 · doi:10.1088/1751-8113/42/10/105205
[8] Aleman, A. and Viola, J., “On weak and strong solution operators for evolution equations coming from quadratic operators,” e-print , 2014. · Zbl 1474.35196
[9] Aleman, A.; Viola, J., Singular-value decomposition of solution operators to model evolution equations, Int. Math. Res. Notices, 2015, 17, 8275-8288 · Zbl 1339.47056 · doi:10.1093/imrn/rnu199
[10] Almog, Y., The stability of the normal state of superconductors in the presence of electric currents, SIAM J. Math. Anal., 40, 824-850 (2008) · Zbl 1165.82029 · doi:10.1137/070699755
[11] Almog, Y.; Helffer, B.; Pan, X.-B., Superconductivity near the normal state in a half-plane under the action of a perpendicular electric current and an induced magnetic field. Part II: The large conductivity limit, SIAM J. Math. Anal., 44, 3671-3733 (2012) · Zbl 1263.82074 · doi:10.1137/110860598
[12] Almog, Y.; Helffer, B.; Pan, X.-B., Superconductivity near the normal state in a half-plane under the action of a perpendicular electric currents and an induced magnetic field, Trans. Am. Math. Soc., 365, 1183-1217 (2013) · Zbl 1267.82139 · doi:10.1090/S0002-9947-2012-05572-3
[13] Bagarello, F., Examples of pseudo-bosons in quantum mechanics, Phys. Lett. A, 374, 3823-3827 (2010) · Zbl 1238.81191 · doi:10.1016/j.physleta.2010.07.044
[14] Bender, C. M., Making sense of non-Hermitian Hamiltonians, Rep. Prog. Phys., 70, 947-1018 (2007) · doi:10.1088/0034-4885/70/6/R03
[15] Bender, C. M.; Boettcher, P. N., Real spectra in non-Hermitian Hamiltonians having \(<mml:math display=''inline`` overflow=''scroll``> <mml:mi mathvariant=''script``>PT\) symmetry, Phys. Rev. Lett., 80, 5243-5246 (1998) · Zbl 0947.81018 · doi:10.1103/PhysRevLett.80.5243
[16] Bender, C. M.; Brody, D. C.; Jones, H. F., Complex extension of quantum mechanics, Phys. Rev. Lett., 89, 270401 (2002) · Zbl 1267.81234 · doi:10.1103/PhysRevLett.89.270401
[17] Blank, J.; Exner, P.; Havlíček, M., Hilbert Space Operators in Quantum Physics (2008) · Zbl 1163.47060
[18] Montrieux, W. B., “Estimation de résolvante et construction de quasimode près du bord du pseudospectre,” e-print [math.SP] (2013).
[19] Borisov, D.; Krejčiřík, D., \(<mml:math display=''inline`` overflow=''scroll``> <mml:mi mathvariant=''script``>PT\)-symmetric waveguides, Integr. Equations Oper. Theory, 62, 489-515 (2008) · Zbl 1178.35141 · doi:10.1007/s00020-008-1634-1
[20] Borisov, D.; Krejčiřík, D., The effective Hamiltonian for thin layers with non-Hermitian Robin-type boundary conditions, Asympt. Anal., 76, 49-59 (2012) · Zbl 1241.47042 · doi:10.3233/ASY-2011-1061
[21] Caliceti, E.; Graffi, S.; Hitrik, M.; Sjöstrand, J., Quadratic \(<mml:math display=''inline`` overflow=''scroll``> <mml:mi mathvariant=''script``>PT\)-symmetric operators with real spectrum and similarity to self-adjoint operators, J. Phys. A: Math. Theor., 45, 444007 (2012) · Zbl 1263.81190 · doi:10.1088/1751-8113/45/44/444007
[22] Davies, E. B., Pseudo-spectra, the harmonic oscillator and complex resonances, Proc. R. Soc. A, 455, 585-599 (1999) · Zbl 0931.70016 · doi:10.1098/rspa.1999.0325
[23] Davies, E. B., Semi-classical states for non-self-adjoint Schrödinger operators, Commun. Math. Phys., 200, 35-41 (1999) · Zbl 0921.47060 · doi:10.1007/s002200050521
[24] Davies, E. B., Wild spectral behaviour of anharmonic oscillators, Bull. London Math. Soc., 32, 432-438 (2000) · Zbl 1043.47502 · doi:10.1112/S0024609300007050
[25] Davies, E. B., Non-self-adjoint differential operators, Bull. London Math. Soc., 34, 513-532 (2002) · Zbl 1052.47042 · doi:10.1112/S0024609302001248
[26] Davies, E. B., Linear Operators and their Spectra (2007) · Zbl 1138.47001
[27] Davies, E. B.; Kuijlaars, A. B. J., Spectral asymptotics of the non-self-adjoint harmonic oscillator, J. London Math. Soc., 70, 420-426 (2004) · Zbl 1073.34093 · doi:10.1112/S0024610704005381
[28] Davies, E. B. and Marletta, M., private communication (2013).
[29] Dencker, N.; Sjöstrand, J.; Zworski, M., Pseudospectra of semiclassical (pseudo-) differential operators, Commun. Pure Appl. Math., 57, 384-415 (2004) · Zbl 1054.35035 · doi:10.1002/cpa.20004
[30] Dieudonné, J., “Quasi-Hermitian operators,” Proceedings of the International Symposium on Linear Spaces (Jerusalem Academic Press, Jerusalem; Pergamon, Oxford,1961), pp. 115-123. · Zbl 0114.31601
[31] Dimassi, M.; Sjöstrand, J., Spectral Asymptotics in the Semi-Classical Limit (1999) · Zbl 0926.35002
[32] Dorey, P.; Dunning, C.; Tateo, R., Spectral equivalences, Bethe Ansatz equations, and reality properties in \(<mml:math display=''inline`` overflow=''scroll``> <mml:mi mathvariant=''script``>PT\)-symmetric quantum mechanics, J. Phys. A: Math. Gen., 34, 5679-5704 (2001) · Zbl 0982.81021 · doi:10.1088/0305-4470/34/28/305
[33] Edmunds, D. E.; Evans, W. D., Spectral Theory and Differential Operators (1987) · Zbl 0628.47017
[34] Fisher, M. E., Yang-Lee edge singularity and \(φ^3\) field theory, Phys. Rev. Lett., 40, 1610-1613 (1978) · doi:10.1103/PhysRevLett.40.1610
[35] Giordanelli, I.; Graf, G. M., The real spectrum of the imaginary cubic oscillator: An expository proof, Ann. Henri Poincaré, 16, 99-112 (2014) · Zbl 1305.81082 · doi:10.1007/s00023-014-0325-5
[36] Graefe, E.-M.; Korsch, H. J.; Rush, A.; Schubert, R., Classical and quantum dynamics in the (non-Hermitian) Swanson oscillator, J. Phys. A: Math. Theor., 48, 055301 (2015) · Zbl 1308.81094 · doi:10.1088/1751-8113/48/5/055301
[37] Hassi, S.; Kuzhel, S., On J-self-adjoint operators with stable C-symmetry, Proc. R. Soc. Edinburgh: Sect. A Math., 143, 141-167 (2013) · Zbl 1400.47003 · doi:10.1017/S0308210511001387
[38] Henry, R., Spectral instability for even non-selfadjoint anharmonic oscillators, J. Spectral Theory, 4, 349-364 (2014) · Zbl 1308.34112 · doi:10.4171/JST/72
[39] Henry, R., Spectral Projections of the Complex Cubic Oscillator, Ann. Henri Poincaré, 15, 2025-2043 (2014) · Zbl 1301.81060 · doi:10.1007/s00023-013-0292-2
[40] Herbst, I. W., Dilation analyticity in constant electric field. I. The two body problem, Commun. Math. Phys., 64, 3, 279-298 (1979) · Zbl 0447.47028 · doi:10.1007/BF01221735
[41] Hitrik, M.; Sjöstrand, J.; Viola, J., Resolvent estimates for elliptic quadratic differential operators, Anal. & PDE, 6, 181-196 (2013) · Zbl 1295.47045 · doi:10.2140/apde.2013.6.181
[42] Hörmander, L., Symplectic classification of quadratic forms, and general Mehler formulas, Math. Z., 219, 3, 413-449 (1995) · Zbl 0829.35150 · doi:10.1007/BF02572374
[43] Hörmander, L., The Analysis of Linear Partial Differential Operators. III. Pseudo-Differential Operators (2007) · Zbl 1115.35005
[44] Kato, T., Perturbation Theory for Linear Operators (1995) · Zbl 0836.47009
[45] Krejčiřík, D., Calculation of the metric in the Hilbert space of a \(<mml:math display=''inline`` overflow=''scroll``> <mml:mi mathvariant=''script``>PT\)-symmetric model via the spectral theorem, J. Phys. A: Math. Theor., 41, 244012 (2008) · Zbl 1142.81008 · doi:10.1088/1751-8113/41/24/244012
[46] Krejčiřík, D.; Siegl, P., \(<mml:math display=''inline`` overflow=''scroll``> <mml:mi mathvariant=''script``>PT\)-symmetric models in curved manifolds, J. Phys. A: Math. Theor., 43, 485204 (2010) · Zbl 1207.81030 · doi:10.1088/1751-8113/43/48/485204
[47] Krejčiřík, D.; Tater, M., Non-Hermitian spectral effects in a \(<mml:math display=''inline`` overflow=''scroll``> <mml:mi mathvariant=''script``>PT\)-symmetric waveguide, J. Phys. A: Math. Theor., 41, 244013 (2008) · Zbl 1140.81386 · doi:10.1088/1751-8113/41/24/244013
[48] Krejčiřík, D.; Bíla, H.; Znojil, M., Closed formula for the metric in the Hilbert space of a \(<mml:math display=''inline`` overflow=''scroll``> <mml:mi mathvariant=''script``>PT\)-symmetric model, J. Phys. A: Math. Gen., 39, 10143-10153 (2006) · Zbl 1117.81058 · doi:10.1088/0305-4470/39/32/S15
[49] Krejčiřík, D.; Siegl, P.; Železný, J., On the similarity of Sturm-Liouville operators with non-Hermitian boundary conditions to self-adjoint and normal operators, Complex Anal. Oper. Theory, 8, 255-281 (2014) · Zbl 1410.34086 · doi:10.1007/s11785-013-0301-y
[50] Mityagin, B.; Siegl, P., Root system of singular perturbations of the harmonic oscillator type operators, Lett. Math. Phys., 2015, 1-21 · Zbl 1330.47023 · doi:10.1007/s11005-015-0805-7
[51] Mityagin, B.; Siegl, P.; Viola, J., Differential operators admitting various rates of spectral projection growth (2013)
[52] Mostafazadeh, A., Pseudo-Hermiticity versus PT symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian, J. Math. Phys., 43, 205-214 (2002) · Zbl 1059.81070 · doi:10.1063/1.1418246
[53] Mostafazadeh, A., Pseudo-Hermiticity versus PT symmetry. II. A complete characterization of non-Hermitian Hamiltonians with a real spectrum, J. Math. Phys., 43, 2814-2816 (2002) · Zbl 1060.81022 · doi:10.1063/1.1461427
[54] Mostafazadeh, A., Pseudo-Hermiticity versus PT symmetry. III. Equivalence of pseudo-Hermiticity and the presence of antilinear symmetries, J. Math. Phys., 43, 3944-3951 (2002) · Zbl 1061.81075 · doi:10.1063/1.1489072
[55] Mostafazadeh, A., Pseudo-Hermitian representation of quantum mechanics, Int. J. Geom. Methods Mod. Phys., 7, 1191-1306 (2010) · Zbl 1208.81095 · doi:10.1142/S0219887810004816
[56] Pravda-Starov, K., A complete study of the pseudo-spectrum for the rotated harmonic oscillator, J. London Math. Soc., 73, 745-761 (2006) · Zbl 1106.34060 · doi:10.1112/S0024610706022952
[57] Pravda-Starov, K., Boundary pseudospectral behaviour for semiclassical operators in one dimension, Int. Math. Res. Not., 2007, 31 · Zbl 1135.47048 · doi:10.1093/imrn/rnm029
[58] Pravda-Starov, K., On the pseudospectrum of elliptic quadratic differential operators, Duke Math. J., 145, 249-279 (2008) · Zbl 1157.35129 · doi:10.1215/00127094-2008-051
[59] Prudnikov, A. P.; Brychkov, Y. A.; Marichev, O. I., Integrals and Series. Special Functions, 2 (1986) · Zbl 0733.00004
[60] Reddy, S. C.; Trefethen, L. N., Pseudospectra of the convection-diffusion operator, SIAM J. Appl. Math., 54, 1634-1649 (1994) · Zbl 0811.35026 · doi:10.1137/S0036139993246982
[61] Reed, M.; Simon, B., Methods of Modern Mathematical Physics. I. Functional Analysis (1972)
[62] Reed, M.; Simon, B., Methods of Modern Mathematical Physics. IV. Analysis of Operators (1978) · Zbl 0401.47001
[63] Reichel, L.; Trefethen, L. N., Eigenvalues and pseudo-eigenvalues of Toeplitz matrices, Linear Algebra Appl., 162-164, 153-185 (1992) · Zbl 0748.15010 · doi:10.1016/0024-3795(92)90374-J
[64] Roch, S.; Silbermann, B., \(C^∗\)-algebra techniques in numerical analysis, J. Oper. Theory, 35, 241-280 (1996) · Zbl 0865.65035
[65] Schechter, M., Operator Methods in Quantum Mechanics (1981) · Zbl 0456.47012
[66] Scholtz, F. G.; Geyer, H. B.; Hahne, F. J. W., Quasi-Hermitian operators in quantum mechanics and the variational principle, Ann. Phys., 213, 74-101 (1992) · Zbl 0749.47041 · doi:10.1016/0003-4916(92)90284-S
[67] Shargorodsky, E., On the definition of pseudospectra, Bull. London Math. Soc., 41, 524-534 (2009) · Zbl 1207.47005 · doi:10.1112/blms/bdp031
[68] Shargorodsky, E., Pseudospectra of semigroup generators, Bull. London Math. Soc., 42, 1031-1034 (2010) · Zbl 1218.47010 · doi:10.1112/blms/bdq062
[69] Shin, K. C., On the reality of the eigenvalues for a class of \(<mml:math display=''inline`` overflow=''scroll``> <mml:mi mathvariant=''script``>PT\)-symmetric oscillators, Commun. Math. Phys., 229, 543-564 (2002) · Zbl 1017.34083 · doi:10.1007/s00220-002-0706-3
[70] Siegl, P., \(<mml:math display=''inline`` overflow=''scroll``> <mml:mi mathvariant=''script``>PT\)-symmetric square well-perturbations and the existence of metric operator, Int. J. Theor. Phys., 50, 991-996 (2011) · Zbl 1216.81068 · doi:10.1007/s10773-010-0593-x
[71] Siegl, P.; Krejčiřík, D., On the metric operator for the imaginary cubic oscillator, Phys. Rev. D, 86, 121702(R) (2012) · doi:10.1103/PhysRevD.86.121702
[72] Simon, B., The bound state of weakly coupled Schrödinger operators in one and two dimensions, Ann. Phys., 97, 279-288 (1976) · Zbl 0325.35029 · doi:10.1016/0003-4916(76)90038-5
[73] Sjöstrand, J., Parametrices for pseudodifferential operators with multiple characteristics, Ark. Mat., 12, 85-130 (1974) · Zbl 0317.35076 · doi:10.1007/BF02384749
[74] Sjöstrand, J., Singularités analytiques microlocales, Astérisque, 95, Volume 95 of Astérisque, 1-166 (1982) · Zbl 0524.35007
[75] Swanson, M. S., Transition elements for a non-Hermitian quadratic Hamiltonian, J. Math. Phys., 45, 585-601 (2004) · Zbl 1070.81053 · doi:10.1063/1.1640796
[76] Szegö, G., Orthogonal Polynomials (1959) · JFM 65.0278.03
[77] Trefethen, L. N.; Embree, M., Spectra and Pseudospectra (2005) · Zbl 1085.15009
[78] Trefethen, L. N.et al., Chebfun Version 4.2. The Chebfun Development Team, 2011, .
[79] Viola, J., Spectral projections and resolvent bounds for partially elliptic quadratic differential operators, J. Pseudo-Differential Oper. Appl., 4, 145-221 (2013) · Zbl 1285.47049 · doi:10.1007/s11868-013-0066-0
[80] Znojil, M., \(<mml:math display=''inline`` overflow=''scroll``> <mml:mi mathvariant=''script``>PT\)-symmetric square well, Phys. Lett. A, 285, 7-10 (2001) · Zbl 0969.81521 · doi:10.1016/S0375-9601(01)00301-2
[81] Zworski, M., A remark on a paper of E. B. Davies, Proc. Am. Math. Soc., 129, 2955-2957 (2001) · Zbl 0981.35107 · doi:10.1090/S0002-9939-01-05909-3
[82] Zworski, M., Semiclassical analysis, Graduate Studies in Mathematics, 138 (2012) · Zbl 1252.58001
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