Scattering and pairing by exchange interactions

Bru, Jean-Bernard  ; Pedra, Walter de Siqueira  ; Santos, Alan Ramer dos 

Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e129

Résumé

Quantum interactions exchanging different types of particles play a pivotal rôle in quantum many-body theory, but they are not sufficiently investigated from a mathematical perspective. Here, we consider a system made of two fermions and one boson, in order to study the effect of such an off-diagonal interaction term, having in mind the physics of cuprate superconductors. Additionally, our model also includes a generalized Hubbard interaction (i.e., a general local repulsion term for the fermions). Regarding pairing, exponentially localized dressed bound fermion pairs are shown to exist, and their effective dispersion relation is studied in detail. Scattering properties of the system are derived for two channels: the unbound and bound pair channels. We give particular attention to the regime of very large on-site (Hubbard) repulsions because this situation is relevant for cuprate superconductors.

Détail

DOI : 10.1017/fms.2025.10083

Bibliographie

[1] Bru, J.-B. and Zagrebnov, V. A., ‘Exact solution of the Bogoliubov Hamiltonian for weakly imperfect Bose gas’, J. Phys. A: Math. Gen. A 31 (1998), 9377–9404.10.1088/0305-4470/31/47/002 Google Scholar | DOI

[2] Bru, J.-B. and Zagrebnov, V. A., ‘Quantum interpretation of thermodynamic behaviour of the Bogoliubov weakly imperfect Bose gas’, Phys. Lett. A 247 (1998), 37–41.10.1016/S0375-9601(98)00530-1 Google Scholar | DOI

[3] Aleksandrov, L., Zagrebnov, V. A., Zh. A. Kozlov, V. A. Parfenov, Priezzhev, V. B., ‘High energy neutron scattering and the Bose condensate in He II’, Sov. Phys.-JETP 41 (1975) 915. Google Scholar

[4] Dokukin, E. V., Kozlov, Zh. K., Parfenov, V. A. and Puchkev, A. V., ‘Investigation of the temperature dependence of the Bose condensate density in helium-4 in relation to the superfluidity phenomenon’, Sov. Phys.-JETP 48 (1978) 1146. Google Scholar

[5] Blagoveshchenskii, N. M., Bogoyavlenskii, I. V., Karnatsevich, L. V., Kolobrodov, V. G., Kozlov, Zh. A., Priezzhev, V. B., Puchkov, A. V., Skomorokhov, A. N. and Yarunin, V. S., ‘Absorption of electromagnetic field energy by the superfluid system of atoms with a dipole moment’, Phys. Rev. B 50 (1994), 16550.10.1103/PhysRevB.50.16550 Google Scholar | DOI

[6] Griffin, A., Snoke, D. W. and Stringari, S. (eds.), Bose-Einstein condensation (Cambridge Univ. Press, Cambridge, 1996). Google Scholar

[7] Griffin, A., Excitations in a Bose-Condensated Liquid (Cambridge Univ. Press, Cambridge, 1993).10.1017/CBO9780511524257 Google Scholar | DOI

[8] Bru, J.-B., ‘Beyond the dilute Bose gas’, Physica A 359 (2006), 306–344.10.1016/j.physa.2005.05.075 Google Scholar | DOI

[9] Lonigro, D., ‘Generalized spin-boson models with non-normalizable form factors’, J. Math. Phys. 63 (2022), 072105.10.1063/5.0085576 Google Scholar | DOI

[10] Bach, V. and Rauch, R., ‘On relative bounds for interacting Fermion operators’, Doc. Math. 28(3) (2023), 683–707.10.4171/dm/919 Google Scholar | DOI

[11] Dutta, O., Gajda, M., Hauke, P., Lewenstein, M., Lühmann, D.-S., Malomed, B. A., Sowiński, T. and Zakrzewski, J., ‘Non-standard Hubbard models in optical lattices: a review’, Rep. Prog. Phys. 78 (2015), 066001 (47pp)10.1088/0034-4885/78/6/066001 Google Scholar PubMed | DOI

[12] Saxena, A. K., High-Temperature Superconductors (Springer-Verlag, Berlin Heidelberg, 2010).10.1007/978-3-642-00712-5 Google Scholar | DOI

[13] Plakida, N., High-Temperature Cuprate Superconductors, Experiment, Theory, and Applications (Springer-Verlag, Berlin Heidelberg, 2010).10.1007/978-3-642-12633-8 Google Scholar | DOI

[14] Wesche, R., Physical Properties of High-Temperature Superconductors (Wiley series in materials for electronic and optoelectronic applications) (John Wiley & Sons, Ltd., Chichester, West Sussex, 2015). Google Scholar

[15] Köppel, H., Yarkony, D. R. and Barentzen, H., The Jahn-Teller Effect: Fundamentals and Implications for Physics and Chemistry(Springer, Berlin Heidelberg, 2009).10.1007/978-3-642-03432-9 Google Scholar | DOI

[16] Müller, K. A. and Bednorz, J. G., ‘Possible high superconductivity in the Ba-La-Cu-O system’, Z. Phys. B: Condens. Matter 64(2) (1986), 189–193. Google Scholar

[17] Müller, K. A., ‘On the superconductivity in hole doped cuprates’, J. Phys.: Condens. Matter 19 (2007), 251002 (13pp). Google Scholar

[18] Keller, H., Bussmann–Holder, A. and Müller, K. A., ‘Jahn–Teller physics and high– superconductivity’, Mater. Today 11(9) (2008), 38–46.10.1016/S1369-7021(08)70178-0 Google Scholar | DOI

[19] Alexandrov, A. S. and Zhao, G. M., ‘Isotope effects in high- cuprate superconductors as support for the bipolaron theory of superconductivity’, New J. Phys. 14 (2012), 013046 (10pp).10.1088/1367-2630/14/1/013046 Google Scholar | DOI

[20] Stoneham, A. M. and Smith, L. W., ‘Defect phenomena in superconducting oxides and analogous ceramic oxides’, J. Phys.: Condens. Matter 3 (1991), 225–278. Google Scholar

[21] Bru, J.-B., De Siqueira Pedra, W. and De Pasquale, A. Delgado, ‘Isotropic Bipolaron-Fermion-Exchange Theory and Unconventional Pairing in Cuprate Superconductors’, Ann. Phys. (Berlin) 531 (2019), 1700235.10.1002/andp.201700235 Google Scholar | DOI

[22] Bru, J.-B., De Siqueira Pedra, W. and De Pasquale, A. Delgado, ‘d-Wave pairing driven by bipolaric modes related to giant electron-phonon anomalies in high- superconductors’, J. Stat. Mech.: Theory Exp. (2015), P03002 (36pp). Google Scholar

[23] Arpaia, R., Martinelli, L., Moretti Sala, M., Caprara, S., Nag, A., Brookes, N. B., Camisa, P., Li, Q., Gao, Q., Zhou, X., Garcia-Fernandez, M., Zhou, K.-J., Schierle, E., Bauch, T., Peng, Y. Y., Di Castro, C., Grilli, M., Lombardi, F., Braicovich, L. and Ghiringhelli, G., ‘Signature of quantum criticality in cuprates by charge density fluctuations’, Nat. Commun. 14 (2023), 7198.10.1038/s41467-023-42961-5 Google Scholar PubMed | DOI

[24] De Pasquale, A. Delgado, Existência de pares “d-wave” e ondas de densidade em uma classe de modelos microscópicos para supercondutores com alta temperatura de transição. PhD thesis, Physics Institute of the University of São Paulo, 2018. DOI:10.11606/T.43.2018.tde-03052018-150652. URL: https://www.teses.usp.br/teses/disponiveis/43/43134/tde-03052018-150652/pt-br.php 10.11606/T.43.2018.tde-03052018-150652 Google Scholar | DOI

[25] Hiroshima, F., Sasaki, I., Spohna, H. and Suzuki, A., ‘Enhanced binding in quantum field theory’, Preprint, 2012, [math-ph]. Google Scholar | arXiv

[26] Galtbayar, A., Jensen, A. and Yajima, K., ‘The Nelson model with less than two photons’, Ann. Henri Poincaré 4 (2003), 239–273.10.1007/s00023-003-0129-5 Google Scholar | DOI

[27] Dayantsolmon, D. and Galtbayar, A., ‘Non-relativistic Pauli-Fierz Hamiltonian for less than two photons’, Hokkaido Math. J. 50(3) (2021), 309–326.10.14492/hokmj/2019-164 Google Scholar | DOI

[28] Olivieri, M., ‘The Casimir-Polder effect for an approximate Pauli-Fierz model: The atom plus wall case’, in Correggi, M. and Falconi, M. (eds.), Quantum Mathematics II. INdAM 2022. (Springer INdAM Series) vol 58 (Springer, Singapore, 2022). Google Scholar

[29] Imada, M., Fujimori, A., and Tokura, Y., ‘Metal-insulator transitions’, Rev. Mod. Phys. 70 (1998), 1039–1263.10.1103/RevModPhys.70.1039 Google Scholar | DOI

[30] Keimer, B., Kivelson, S. A., Norman, M. R., Uchida, S. and Zaanen, J., ‘From quantum matter to high-temperature superconductivity in copper oxides’, Nature 518 (2015), 179–186.10.1038/nature14165 Google Scholar PubMed | DOI

[31] Tsuei, C. C. and Kirtley, J. R., ‘Pairing symmetry in cuprate superconductors: Phase-sensitive tests, in Bennemann, K.-H. and Ketterson, J. B. (eds.), The Physics of Superconductors: Vol. I. Conventional and High-Tc Superconductors (Springer-Verlag, Berlin Heidelberg, 2003). Google Scholar

[32] Lagoin, C., Bhattacharya, U., Grass, T., Chhajlany, R. W., Salamon, T., Baldwin, K., Pfeiffer, L., Lewenstein, M., Holzmann, M. and Dubin, F., ‘Extended Bose-Hubbard model with dipolar excitons’, Nature 609 (2022), 485–489.10.1038/s41586-022-05123-z Google Scholar PubMed | DOI

[33] Tsuei, C. C. and Kirtley, J. R., ‘Pairing symmetry in cuprate superconductors’, Rev. Mod. Phys. 72(4) (2000), 969–1016.10.1103/RevModPhys.72.969 Google Scholar | DOI

[34] Dzhumanov, S., Theory of Conventional and Unconventional Superconductivity in the High- Cuprates and Other Systems (Nova Science Publishers, Inc, New York, 2013). Google Scholar

[35] Dzhumanov, S., ‘Possible insulating, metallic and superconducting states in doped high- superconductors’, Solid State Commun. 115 (2000), 155–160.10.1016/S0038-1098(00)00052-1 Google Scholar | DOI

[36] Dzhumanov, S., Karimboev, E. X. and Djumanov, Sh. S., ‘Underlying mechanisms of pseudogap phenomena and Bose-liquid superconductivity in high- cuprates’, Phys. Lett. A 380 (2016), 2173–2180.10.1016/j.physleta.2016.04.038 Google Scholar | DOI

[37] Reagor, D., ‘Large dielectric constants and massive carriers in ’, Phys. Rev. Lett. 62(17) (1989), 2048–2051.10.1103/PhysRevLett.62.2048 Google Scholar | DOI

[38] Reed, M. and Simon, B., Methods of Modern Mathematical Physics, Vol. IV: Analysis of Operators (Academic Press, London, 1978). Google Scholar

[39] Höck, K. H., Nickisch, H. and Thomas, H., ‘Jahn-Teller effect in itinerant electron systems: The Jahn-Teller polaron’, Helv. Phys. Acta 56 (1983), 237–243. Google Scholar

[40] Kato, T.. Perturbation Theory for Linear Operators, second edn. (Springer-Verlag, Berlin–Heidelberg–New-York, 1980). Google Scholar

[41] Reed, M. and Simon, B., Methods of Modern Mathematical Physics, Vol. III: Scattering Theory (Academic Press, New York-London, 1979). Google Scholar

[42] Newton, R. G., Scattering Theory of Waves and Particles, second edn. (Springer, New York, 1982).10.1007/978-3-642-88128-2 Google Scholar | DOI

[43] Yafaev, D., Scattering Theory: Some Old and New Problems (Lecture Notes Math.) vol. 1735 (Springer-Verlag, 2000).10.1007/BFb0105531 Google Scholar | DOI

[44] Rudin, W., Functional Analysis (McGraw-Hill Science, New York, 1991). Google Scholar

[45] Nielsen, O. A., Direct Integral Theory (Lecture Notes in Pure and Applied Mathematics) vol. 61 (Marcel Dekker, New York and Basel, 1980). Google Scholar

[46] Bru, J.-B. and Pedra, De Siqueira, -Algebra and Mathematical Foundations of Quantum Statistical Mechanics (Latin American Mathematics Series - UFSCar subseries) (Springer Nature Switzerland AG, 2023).10.1007/978-3-031-28949-1 Google Scholar | DOI

[47] Combes, J. M. and Thomas, L., ‘Asymptotic behaviour of eigenfunctions for multiparticle Schrödinger operators’, Comm. Math. Phys. 34(4) (1973), 251–270.10.1007/BF01646473 Google Scholar | DOI

[48] Reed, M. and Simon, B., Methods of Modern Mathematical Physics, Vol. II: Fourier Analysis, Self-Adjointness (Academic Press, New York-London, 1975). Google Scholar

[49] Mityagin, B., ‘The zero set of a real analytic function’, Preprint, 2015, [math.CA]. Google Scholar | arXiv

[50] Folland, G. B., A Course in Abstract Harmonic Analysis, second edn. (Chapman and Hall/CRC, New York, 2015). Google Scholar

[51] Kittel, C., Introduction to Solid State Physics, eighth edn. (Wiley, Hoboken, New Jersey, 2005). Google Scholar

[52] Chen, C. Y., Birgeneau, R. J., Kastner, M. A., Preyer, N. W. and Thio, T., ‘Frequency and magnetic-field dependence of the dielectric constant and conductivity of ’, Phys. Rev. B 43(1) (1991), 392–401.10.1103/PhysRevB.43.392 Google Scholar | DOI

[53] Božović, I., He, X., Wu, J. and Bollinger, A. T., ‘Dependence of the critical temperature in overdoped copper oxides on superfluid density’, Nature 536 (2016), 309–311.10.1038/nature19061 Google Scholar PubMed | DOI

[54] Rodgers, P., ‘Superconductivity debate gets ugly’, vol. 11 (Physics World, 1998), 15–16. Google Scholar

[55] Ranninger, J. and Robaszkiewicz, S., ‘Superconductivity of locally paired electrons’, Physica B+C 135, (1985), 468–472.10.1016/0378-4363(85)90533-9 Google Scholar | DOI

[56] Ranninger, J. and Robin, J. M., ‘The boson-fermion model of high-Tc superconductivity’, Doping dependence, Phys. C (Amsterdam, Neth.) 253(3–4) (1995), 279–291.10.1016/0921-4534(95)00515-3 Google Scholar | DOI

[57] Ranninger, J., ‘The polaron scenario for high- superconductivity’, Phys. C (Amsterdam, Neth.) 235–240 (1994), 277–280.10.1016/0921-4534(94)91368-4 Google Scholar | DOI

[58] Ionov, S. P., ‘Paired electron processes of localization-delocalization in condensed media’, Izv. Akad. Nauk 49 (1985), 310; English translation: Bull. Acad. Sci. USSR, Phys. Ser. (USA) A (1985), 90. Google Scholar

[59] Müller, K. A., ‘The polaronic basis for high-temperature superconductivity’, J Supercond Nov Magn 30 (2017), 3007–3018.10.1007/s10948-017-4262-7 Google Scholar | DOI

[60] Bianconi, A., Castellano, A. Congiu, De Santis, M., Rudolf, P., Lagarde, P., Flank, A. M. and Marcelli, A., ‘L xanes of the high Tc superconductor YBaCuO with variable oxygen content’, Solid State Commun. 63(11), (1987), 1009–1013.10.1016/0038-1098(87)90650-8 Google Scholar | DOI

[61] Bianconi, A., Clozza, A., Castellano, A. Congiu, Longa, S. Della, De Santis, M., Di Cicco, A., Garg, K., Delogu, P., Gargano, A., Giorgi, R., Lagarde, P., Flank, A. M. and Marcelli, A., ‘Experimental evidence of itinerant Cu 3d - Oxygen-hole many body configuration in the High-Tc superconductor YBaCuO’, Int. J. Mod. Phys. B 1(3–4) (1987), 853–862.10.1142/S0217979287001213 Google Scholar | DOI

[62] Bianconi, A., Clozza, A., Castellano, A. Congiu, Longa, S. Della, De Santis, M., Di Cicco, A., Garg, K., Delogu, P., Gargano, A., Giorgi, R., Lagarde, P., Flank, A. M. and Marcelli, A., ‘Cu 3d - Ligand hole configuration in YBaCuO by X-ray spectroscopies’, J. Phys. Colloques 48(C9) (1987), C9-1179–C9-1184.10.1051/jphyscol:19879212 Google Scholar | DOI

[63] Bianconi, A., Budnick, J., Flank, A. M., Fontaine, A., Lagarde, P., Marcelli, A., Tolentino, H., Chamberland, B., Michel, C., Raveau, B. and Demazeau, G., ‘Evidence of 3d -ligand hole states in the superconductor from L X-ray absorption spectroscopy’, Phys. Lett. A 127(5) (1988), 285–291.10.1016/0375-9601(88)90698-6 Google Scholar | DOI

[64] Bianconi, A., Castellano, A. Congiu, De Santis, M., Delogu, P., Gargano, A. and Giorgi, R., ‘Localization of Cu 3d levels in the high Tc superconductor YBaCuO by Cu 2p X-ray photoelectron spectroscopy’, Solid State Commun. 63(12) (1987), 1135–1139.10.1016/0038-1098(87)91063-5 Google Scholar | DOI

[65] Fujimori, A., Takayama-Muromachi, E. and Uchida, Y., ‘Electronic structure of superconducting Cu oxides’, Solid State Commun. 63(9) (1987), 857–860.10.1016/0038-1098(87)90901-X Google Scholar | DOI

[66] Bianconi, A., ‘Lifshitz transitions in multi-band Hubbard models for topological superconductivity in complex quantum matter’, J. Supercond. Novel Magn. 31 (2018), 603–610.10.1007/s10948-017-4535-1 Google Scholar | DOI

[67] Dean, M. P. M., Dellea, G., Springell, R. S., Yakhou-Harris, F., Kummer, K., Brookes, N. B., Liu, X., Sun, Y.-J., Strle, J., Schmitt, T., Braicovich, L., Ghiringhelli, G., Božović, I. and Hill, J. P. ‘Persistence of magnetic excitations in from the undoped insulator to the heavily overdoped non-superconducting metal’, Nat. Mater. 12 (2013), 1019–1023.10.1038/nmat3723 Google Scholar | DOI

[68] Le Tacon, M., Ghiringhelli, G., Chaloupka, J., Sala, M. M., Hinkov, V., Haverkort, M. W., Minola, M., Bakr, M., Zhou, K. J., Blanco-Canosa, S., Monney, C., Song, Y. T., Sun, G. L., Lin, C. T., De Luca, G. M., Salluzzo, M., Khaliullin, G., Schmitt, T., Braicovich, L. and Keimer, B. ‘Intense paramagnon excitations in a large family of high-temperature superconductors’, Nat Phys 7 (2011), 725–730.10.1038/nphys2041 Google Scholar | DOI

[69] Mihailovic, D., ‘Optical experimental evidence for a universal length scale for the dynamic charge inhomogeneity of cuprate superconductors’, Phys. Rev. Lett. 94 (2005), 207001 (4pp).10.1103/PhysRevLett.94.207001 Google Scholar PubMed | DOI

[70] Hague, J. P., Kornilovitch, P. E., Samson, J. H. and Alexandrov, A. S., ‘Superlight small bipolarons in the presence of a strong Coulomb repulsion’, Phys. Rev. Lett. 98 (2007), 037002 (4pp).10.1103/PhysRevLett.98.037002 Google Scholar PubMed | DOI

[71] Alexandrov, A. S., ‘Bose–Einstein condensation of strongly correlated electrons and phonons in cuprate superconductors’, J. Phys.: Condens. Matter 19 (2007), 125216 (23pp). Google Scholar

[72] Alexandrov, A. S., ‘Theory of high-temperature superconductivity in doped polar insulators’, EPL 95 (2011), 27004 (5pp).10.1209/0295-5075/95/27004 Google Scholar | DOI

[73] Alexandrov, A. S., ‘High-temperature superconductivity: the explanation’, Phys. Scr. 83 (2011), 038301 (8pp).10.1088/0031-8949/83/03/038301 Google Scholar | DOI

[74] Alexandrov, A. S., ‘Theory of high temperature superconductivity beyond BCS with realistic Coulomb and Fröhlich interactions’, J. Supercond. Novel Magn. 26(4) (2013), 1313–1317.10.1007/s10948-012-2098-8 Google Scholar | DOI

[75] Padilla, W. J., Lee, Y. S., Dumm, M., Blumberg, G., Ono, S., Segawa, K., Komiya, S., Ando, Y. and Basov, D. N., ‘Constant effective mass across the phase diagram of high-cuprates’, Phys. Rev. B 72 (2005), 060511(R) (4pp). Google Scholar

[76] Pintschovius, L. and Reichardt, W., ‘Phonon dispersions and phonon density-of-states in copper-oxide superconductors’, in Furrer, A. (ed.), Neutron Scattering in Layered Copper-Oxide Superconductors (Physics and Chemistry of Materials with Low-Dimensional Structures) vol. 20 (Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998), 165–223.10.1007/978-94-015-1284-8_5 Google Scholar | DOI

[77] Pintschovius, L., Reznik, D. and Yamada, K., ‘Oxygen phonon branches in overdoped ’. Phys. Rev. B 74 (2006), 174514 (5pp).10.1103/PhysRevB.74.174514 Google Scholar | DOI

[78] Reznik, D., Pintschovius, L., Fujita, M., Yamada, K., Gu, G. D and Tranquada, J. M., ‘Electron-phonon anomaly related to charge stripes: static stripe phase versus optimally doped superconducting ’, J. Low Temp. Phys. 147(3–4) (2007), 353–364.10.1007/s10909-007-9318-9 Google Scholar | DOI

[79] Reznik, D., ‘Giant electron–phonon anomaly in doped and others cuprates’, Adv. Condens. Matter Phys. 2010 (2010), 523549 (24pp). Google Scholar

[80] Reznik, D., Sangiovanni, G., Gunnarsson, O. and Devereaux, T. P., ‘Photoemission kinks and phonons in cuprates’, Nature 455(7213) (2008), E6–E7.10.1038/nature07364 Google Scholar PubMed | DOI

[81] Chaibi, W., Peláez, R. J., Blondel, C., Drag, C. and Delsart, C., ‘Effect of a magnetic field in photodetachment microscopy’, Eur. Phys. J. D 58 (2010), 29–37.10.1140/epjd/e2010-00086-7 Google Scholar | DOI

[82] Giuliani, G. F. and Vignale, G., Quantum Theory of the Electron Liquid (Cambridge, Cambrigde Univ. Press, 2005).10.1017/CBO9780511619915 Google Scholar | DOI

[83] Kabanov, V. V. and Mihailovic, D., ‘Finite-wave-vector phonon coupling to degenerate electronic states in ’, J. Supercond. Novel Magn. 13 (2000), 959–962.10.1023/A:1026450207961 Google Scholar | DOI

[84] D. Mihailovic D and Kabanov, V. V., ‘Finite wave vector Jahn-Teller pairing and superconductivity in the cuprates’, Phys. Rev. B 63 (2001), 054505 (8pp). Google Scholar

[85] Timusk, T. and Statt, B., ‘The pseudogap in high-temperature superconductors: An experimental survey’, Rep. Prog. Phys. 62 (1999), 61–122.10.1088/0034-4885/62/1/002 Google Scholar | DOI

[86] Panson, A. J., Wagner, G. R., Braginski, A. I., Gavaler, J. R., Janocko, M. A., Pohl, H. C. and Talvacchio, J., ‘Properties of superconductors’, Appl. Phys. Lett. 50 (1987) 1104–1106.10.1063/1.97932 Google Scholar | DOI

[87] Müller, K. A., Zhao, G. M., Conder, K. and Keller, H., ‘The ratio of small polarons to free carriers in derived from susceptibility measurements’, J. Phys.: Condens. Matter 10(18) (1998), L291–L296. Google Scholar

[88] Mourachkine, A., High-Temperature Superconductivity in Cuprates: The Nonlinear Mechanism and Tunneling Measurements (Kluwer Academic Publishers, Dordrecht, 2002).10.1007/0-306-48063-8 Google Scholar | DOI

[89] Hwang, J., ‘Superconducting coherence length of hole-doped cuprates obtained from electron–boson spectral density function’, Sci. Rep. 11 (2021), 11668 (7pp).10.1038/s41598-021-91163-w Google Scholar PubMed | DOI

[90] Bratteli, O. and Robinson, D. W., Operator Algebras and Quantum Statistical Mechanics, vol. II, second edn. (Springer-Verlag, New York, 1997).10.1007/978-3-662-03444-6 Google Scholar | DOI

[91] De Oliveira, O. R. Branco, ‘The Implicit and the Inverse Function theorems: easy proofs’, Preprint, 2012, [math.CA]. Google Scholar | arXiv

[92] Toland, J., Lectures on Real-Analytic Operator Equations, http://www.dma.unina.it/hamiltonianPDE/mate/tolandCapri.pdf. Google Scholar

[93] Rudin, W., Principles of mathematical Analysis (McGraw Hill, New York, 1953). Google Scholar

[94] Folland, G. B., Real Analysis, Modern Techniques and Their Applications, second edn. (John Wiley & Sons, New York, 1999). Google Scholar

[95] Schmüdgen, K., Unbounded Self-Adjoint Operators on Hilbert Space (Graduate Texts in Mathematics) (Springer, Dordrecht, 2012).10.1007/978-94-007-4753-1 Google Scholar | DOI

[96] Kharazishvili, A., Notes on Real Analysis and Measure Theory (Springer Monographs in Mathematics, Springer Cham, 2023). Google Scholar