{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,11]],"date-time":"2026-02-11T14:06:37Z","timestamp":1770818797850,"version":"3.50.1"},"reference-count":45,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2022,7,22]],"date-time":"2022-07-22T00:00:00Z","timestamp":1658448000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,7,22]],"date-time":"2022-07-22T00:00:00Z","timestamp":1658448000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"National Science Center, Poland","award":["2017\/27\/B\/ST3\/02881"],"award-info":[{"award-number":["2017\/27\/B\/ST3\/02881"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Sci Rep"],"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We study the role of disorder in the vibration spectra of molecules and atoms in solids. This disorder may be described phenomenologically by a fractional generalization of ordinary quantum-mechanical oscillator problem. To be specific, this is accomplished by the introduction of a so-called fractional Laplacian (Riesz fractional derivative) to the Scr\u00f6dinger equation with three-dimensional (3D) quadratic potential. To solve the obtained 3D spectral problem, we pass to the momentum space, where the problem simplifies greatly as fractional Laplacian becomes simply<jats:inline-formula><jats:alternatives><jats:tex-math>$$k^\\mu $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msup><mml:mi>k<\/mml:mi><mml:mi>\u03bc<\/mml:mi><\/mml:msup><\/mml:math><\/jats:alternatives><\/jats:inline-formula>,<jats:italic>k<\/jats:italic>is a modulus of the momentum vector and<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mu $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>\u03bc<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>is L\u00e9vy index, characterizing the degree of disorder. In this case,<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mu \\rightarrow 0$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>\u03bc<\/mml:mi><mml:mo>\u2192<\/mml:mo><mml:mn>0<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>corresponds to the strongest disorder, while<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mu \\rightarrow 2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>\u03bc<\/mml:mi><mml:mo>\u2192<\/mml:mo><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>to the weakest so that the case<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mu =2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>\u03bc<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>corresponds to \u201cordinary\u201d (i.e. that without fractional derivatives) 3D quantum harmonic oscillator. Combining analytical (variational) and numerical methods, we have shown that in the fractional (disordered) 3D oscillator problem, the famous orbital momentum degeneracy is lifted so that its energy starts to depend on orbital quantum number<jats:italic>l<\/jats:italic>. These features can have a strong impact on the physical properties of many solids, ranging from multiferroics to oxide heterostructures, which, in turn, are usable in modern microelectronic devices.<\/jats:p>","DOI":"10.1038\/s41598-022-16597-2","type":"journal-article","created":{"date-parts":[[2022,7,22]],"date-time":"2022-07-22T12:04:29Z","timestamp":1658491469000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Fractional quantum oscillator and disorder in the vibrational spectra"],"prefix":"10.1038","volume":"12","author":[{"given":"V. A.","family":"Stephanovich","sequence":"first","affiliation":[]},{"given":"E. V.","family":"Kirichenko","sequence":"additional","affiliation":[]},{"given":"V. K.","family":"Dugaev","sequence":"additional","affiliation":[]},{"given":"Jackie Harjani","family":"Sauco","sequence":"additional","affiliation":[]},{"given":"Bel\u00e9n L\u00f3pez","family":"Brito","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,7,22]]},"reference":[{"key":"16597_CR1","doi-asserted-by":"publisher","first-page":"3623","DOI":"10.1021\/jz4020162","volume":"4","author":"HJ Snaith","year":"2013","unstructured":"Snaith, H. J. Perovskites: The emergence of a new era for low-cost, high-efficiency solar cells. J. Phys. Chem. Lett. 4, 3623 (2013).","journal-title":"J. Phys. Chem. Lett."},{"key":"16597_CR2","doi-asserted-by":"publisher","first-page":"391","DOI":"10.1038\/nnano.2015.90","volume":"10","author":"SD Stranks","year":"2015","unstructured":"Stranks, S. D. & Snaith, H. J. Metal-halideperovskites for photovoltaic and light-emitting devices. Nat. Nanotechnol. 10, 391 (2015).","journal-title":"Nat. Nanotechnol."},{"key":"16597_CR3","doi-asserted-by":"publisher","first-page":"636","DOI":"10.1038\/nmat4271","volume":"14","author":"H Zhu","year":"2015","unstructured":"Zhu, H. et al. Lead halide perovskite nanowire lasers with low lasing thresholds and high quality factors. Nat. Mater. 14, 636 (2015).","journal-title":"Nat. Mater."},{"key":"16597_CR4","doi-asserted-by":"publisher","first-page":"476","DOI":"10.1038\/nmat3911","volume":"13","author":"G Xing","year":"2014","unstructured":"Xing, G. et al. Low-temperature solution-processed wavelength-tunable perovskites for lasing. Nat. Mater. 13, 476 (2014).","journal-title":"Nat. Mater."},{"key":"16597_CR5","doi-asserted-by":"publisher","first-page":"519","DOI":"10.1080\/00018732.2015.1114338","volume":"64","author":"S Dong","year":"2015","unstructured":"Dong, S., Liu, J.-M., Cheong, S.-W. & Ren, Z. Multiferroic materials and magnetoelectric physics: Symmetry, entanglement, excitation, and topology. Adv. Phys. 64, 519 (2015).","journal-title":"Adv. Phys."},{"key":"16597_CR6","volume-title":"Fractional Integrals and Derivatives","author":"SG Samko","year":"2003","unstructured":"Samko, S. G., Kilbas, A. A. & Maritchev, O. I. Fractional Integrals and Derivatives (Gordon and Breach, 2003)."},{"key":"16597_CR7","volume-title":"Fractional Differential Equations","author":"I Podlubny","year":"1999","unstructured":"Podlubny, I. Fractional Differential Equations (Academic Press, 1999)."},{"key":"16597_CR8","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/S0370-1573(00)00070-3","volume":"339","author":"R Metzler","year":"2000","unstructured":"Metzler, R. & Klafter, J. The random walk\u2019s guide to anomalous diffusion a fractional dynamics approach. Phys. Rep. 339, 1 (2000).","journal-title":"Phys. Rep."},{"key":"16597_CR9","doi-asserted-by":"publisher","first-page":"R161","DOI":"10.1088\/0305-4470\/37\/31\/R01","volume":"37","author":"R Metzler","year":"2004","unstructured":"Metzler, R. & Klafter, J. The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A Math. Gen. 37, R161 (2004).","journal-title":"J. Phys. A Math. Gen."},{"key":"16597_CR10","volume-title":"Th\u00e9orie de l\u2019Addition des Variables al\u00e9atoires","author":"P L\u00e9vy","year":"1954","unstructured":"L\u00e9vy, P. Th\u00e9orie de l\u2019Addition des Variables al\u00e9atoires (Gauthier-Villars, 1954)."},{"key":"16597_CR11","unstructured":"L\u00e9vy Flights and related topics in physics (eds. Shlesinger, M. F., Zaslavsky, G. M. & Frisch, U. ). in Lecture Notes in Physics. (Springer, 1995)."},{"key":"16597_CR12","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198537885.001.0001","volume-title":"Random Walks and Random Environments","author":"BD Hughes","year":"1995","unstructured":"Hughes, B. D. Random Walks and Random Environments Vol. 1 (Clarendon Press, 1995)."},{"key":"16597_CR13","volume-title":"Stable Non-Gaussian Random Processes","author":"G Samorodnitsky","year":"1994","unstructured":"Samorodnitsky, G. & Taqqu, M. S. Stable Non-Gaussian Random Processes (Chapman and Hall, 1994)."},{"key":"16597_CR14","unstructured":"Handbook of Mathematical Functions (eds. Abramowitz, M. & Stegun, I.) (Dover, 1972)."},{"key":"16597_CR15","doi-asserted-by":"publisher","first-page":"2736","DOI":"10.1103\/PhysRevE.59.2736","volume":"59","author":"S Jespersen","year":"1999","unstructured":"Jespersen, S., Metzler, R. & Fogedby, H. C. L\u00e9vy flights in external force fields: Langevin and fractional Fokker-Planck equations and their solutions. Phys. Rev. E 59, 2736 (1999).","journal-title":"Phys. Rev. E"},{"key":"16597_CR16","doi-asserted-by":"publisher","first-page":"3135","DOI":"10.1103\/PhysRevE.62.3135","volume":"62","author":"N Laskin","year":"2000","unstructured":"Laskin, N. Fractional quantum mechanics. Phys. Rev. E 62, 3135 (2000).","journal-title":"Phys. Rev. E"},{"key":"16597_CR17","doi-asserted-by":"publisher","DOI":"10.1142\/10541","volume-title":"Fractional Quantum Mechanics","author":"N Laskin","year":"2018","unstructured":"Laskin, N. Fractional Quantum Mechanics (World Scientific, 2018)."},{"key":"16597_CR18","volume-title":"Quantum Mechanics. Nonrelativistic Theory","author":"LD Landau","year":"1995","unstructured":"Landau, L. D. & Lifshits, E. M. Quantum Mechanics. Nonrelativistic Theory (Pergamon Press, 1995)."},{"key":"16597_CR19","volume-title":"Practical Quantum Mechanics","author":"S Fl\u00fcgge","year":"1999","unstructured":"Fl\u00fcgge, S. Practical Quantum Mechanics (Springer, 1999)."},{"key":"16597_CR20","doi-asserted-by":"publisher","first-page":"052127","DOI":"10.1103\/PhysRevE.98.052127","volume":"98","author":"EV Kirichenko","year":"2018","unstructured":"Kirichenko, E. V. & Stephanovich, V. A. Confinement of L\u00e9vy flights in a parabolic potential and fractional quantum oscillator. Phys. Rev. E 98, 052127 (2018).","journal-title":"Phys. Rev. E"},{"key":"16597_CR21","volume-title":"Handbook of Exact Solutions of Ordinary Differential Equations","author":"AD Polyanin","year":"1995","unstructured":"Polyanin, A. D. & Zaitsev, V. F. Handbook of Exact Solutions of Ordinary Differential Equations (CRC Press, 1995)."},{"key":"16597_CR22","volume-title":"An Introduction to Analysis","author":"JR Kirkwood","year":"1995","unstructured":"Kirkwood, J. R. An Introduction to Analysis (PWS Publishing, 1995)."},{"key":"16597_CR23","doi-asserted-by":"publisher","first-page":"11956","DOI":"10.1038\/s41598-021-91414-w","volume":"11","author":"EV Kirichenko","year":"2021","unstructured":"Kirichenko, E. V. & Stephanovich, V. A. The influence of Coulomb interaction screening on the excitons in disordered two-dimensional insulators. Sci. Rep. 11, 11956. https:\/\/doi.org\/10.1038\/s41598-021-91414-w (2021).","journal-title":"Sci. Rep."},{"key":"16597_CR24","doi-asserted-by":"publisher","first-page":"21847","DOI":"10.1039\/C9CP04111G","volume":"21","author":"EV Kirichenko","year":"2019","unstructured":"Kirichenko, E. V. & Stephanovich, V. A. The influence of disorder on the exciton spectra in two-dimensional structures. Phys. Chem. Chem. Phys. 21, 21847 (2019).","journal-title":"Phys. Chem. Chem. Phys."},{"key":"16597_CR25","doi-asserted-by":"publisher","first-page":"24462","DOI":"10.1039\/D0CP03055D","volume":"22","author":"VA Stephanovich","year":"2020","unstructured":"Stephanovich, V. A. & Olchawa, W. L\u00e9vy distributions and disorder in excitonic spectra. Phys. Chem. Chem. Phys. 22, 24462 (2020).","journal-title":"Phys. Chem. Chem. Phys."},{"key":"16597_CR26","doi-asserted-by":"publisher","first-page":"1492","DOI":"10.1103\/PhysRev.109.1492","volume":"109","author":"PW Anderson","year":"1958","unstructured":"Anderson, P. W. Absence of diffusion in certain random lattices. Phys. Rev. 109, 1492 (1958).","journal-title":"Phys. Rev."},{"key":"16597_CR27","volume-title":"Solid State Physics","author":"NW Ashkroft","year":"1976","unstructured":"Ashkroft, N. W. & Mermin, N. D. Solid State Physics (Harcourt, 1976)."},{"key":"16597_CR28","unstructured":"Laguta, V. V., Stephanovich, V. A., Raevski, I. P., Raevskaya, S. I., Titov, V.V., Smotrakov, V.G., & Eremkin, V.V. Magnetoelectric effect in antiferromagnetic multiferroic Pb$$({{\\rm Fe}}_{1\/2}{{\\rm Nb}}_{1\/2}){{\\rm O}}_3$$ and its solid solutions with $${{\\rm PbTiO}}_3$$. Phys. Rev. B 95, 014207 (2017)."},{"key":"16597_CR29","doi-asserted-by":"publisher","first-page":"7229","DOI":"10.1039\/C6CP00054A","volume":"18","author":"VA Stephanovich","year":"2016","unstructured":"Stephanovich, V. A. & Laguta, V. V. Transversal spin freezing and re-entrant spin glass phases in chemically disordered Fe-containing perovskite multiferroics. Phys. Chem. Chem. Phys. 18, 7229 (2016).","journal-title":"Phys. Chem. Chem. Phys."},{"key":"16597_CR30","unstructured":"Zagorodniy, Yu. O., Kuzian, R. O., Kondakova, I. V., Mary\u0161ko, M., Chlan, V., \u0160t\u011bp\u00e1nkov\u00e1, H., Olekhnovich, N. M., Pushkarev, A. V., Radyush, Yu. V., Raevski, I. P., Zalar, B., Laguta, V. V., & Stephanovich, V. A. Chemical disorder and $$^{207}$$Pb hyperfine fields in the magnetoelectric multiferroic Pb$$({\\text{Fe}}_{1\/2}{\\text{ Sb }}_{1\/2}){\\text{ O }}_3$$ and its solid solution with Pb$$({\\text{ Fe }}_{1\/2}{\\text{ Nb }}_{1\/2}){\\text{ O }}_3$$. Phys. Rev. Mater. 2, 014401 (2018)."},{"key":"16597_CR31","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-61544-3_4","volume-title":"The Fokker-Planck Equation","author":"H Risken","year":"1996","unstructured":"Risken, H. The Fokker-Planck Equation (Springer, 1996)."},{"key":"16597_CR32","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4939-1323-7","volume-title":"Stochastic Processes and Applications","author":"GA Pavliotis","year":"2014","unstructured":"Pavliotis, G. A. Stochastic Processes and Applications (Springer, 2014)."},{"key":"16597_CR33","volume-title":"The Transition to Chaos. Conservative Classical Systems and Quantum Manifestations","author":"LE Reichl","year":"2004","unstructured":"Reichl, L. E. The Transition to Chaos. Conservative Classical Systems and Quantum Manifestations Vol. 2 (Springer, 2004)."},{"key":"16597_CR34","doi-asserted-by":"publisher","first-page":"013624","DOI":"10.1103\/PhysRevA.87.013624","volume":"87","author":"J Larson","year":"2013","unstructured":"Larson, J., Anderson, B. M. & Altland, A. Chaos-driven dynamics in spin-orbit-coupled atomic gases. Phys. Rev. A 87, 013624 (2013).","journal-title":"Phys. Rev. A"},{"key":"16597_CR35","first-page":"78","volume":"39","author":"YuA Bychkov","year":"1984","unstructured":"Bychkov, Yu. A. & Rashba, E. I. Properties of a 2D electron gas with lifted spectral degeneracy. JETP Lett. 39, 78 (1984).","journal-title":"JETP Lett."},{"key":"16597_CR36","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1002\/andp.202000012","volume":"2000012","author":"EV Kirichenko","year":"2020","unstructured":"Kirichenko, E. V., Stephanovich, V. A. & Sherman, E. Y. Chaotic cyclotron and Hall trajectories due to spin-orbit coupling. Ann. Phys. (Berlin) 2000012, 1\u20138. https:\/\/doi.org\/10.1002\/andp.202000012 (2020).","journal-title":"Ann. Phys. (Berlin)"},{"key":"16597_CR37","doi-asserted-by":"crossref","unstructured":"Stephanovich, V.A., Sherman, E. Ya., Zinner, N.T., &Marchukov, O.V. Energy-level repulsion by spin-orbit coupling in two-dimensional Rydberg excitons. Phys. Rev. B 97, 205407 (2018).","DOI":"10.1103\/PhysRevB.97.205407"},{"key":"16597_CR38","doi-asserted-by":"publisher","first-page":"2895","DOI":"10.1021\/acs.nanolett.7b00064","volume":"17","author":"M Fu","year":"2017","unstructured":"Fu, M. et al. Neutral and charged exciton fine structure in single lead halide perovskite nanocrystals revealed by magneto-optical spectroscopy. Nano Lett. 17, 2895 (2017).","journal-title":"Nano Lett."},{"key":"16597_CR39","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1088\/0034-4885\/49\/1\/001","volume":"49","author":"K Berkelman","year":"1986","unstructured":"Berkelman, K. Heavy quark physics. Rep. Prog. Phys. 49, 1 (1986).","journal-title":"Rep. Prog. Phys."},{"key":"16597_CR40","doi-asserted-by":"publisher","first-page":"4419","DOI":"10.1016\/j.physa.2010.06.036","volume":"389","author":"P Garbaczewski","year":"2010","unstructured":"Garbaczewski, P. & Stephanovich, V. L\u00e9vy flights in inhomogeneous environments. Physica A 389, 4419 (2010).","journal-title":"Physica A"},{"key":"16597_CR41","doi-asserted-by":"publisher","first-page":"31","DOI":"10.3390\/math4020031","volume":"4","author":"A Liemert","year":"2016","unstructured":"Liemert, A. & Kienle, A. Fractional Schr\u00f6dinger equation in the presence of the linear potential. Mathematics 4, 31. https:\/\/doi.org\/10.3390\/math4020031 (2016).","journal-title":"Mathematics"},{"key":"16597_CR42","doi-asserted-by":"publisher","first-page":"072103","DOI":"10.1063\/1.4814049","volume":"54","author":"P Garbaczewski","year":"2013","unstructured":"Garbaczewski, P. & Stephanovich, V. L\u00e9vy flights and nonlocal quantum dynamics. J. Math. Phys. 54, 072103 (2013).","journal-title":"J. Math. Phys."},{"key":"16597_CR43","doi-asserted-by":"publisher","first-page":"981","DOI":"10.1088\/0954-3899\/26\/6\/401","volume":"26","author":"RL Hall","year":"2000","unstructured":"Hall, R. L. A simple interpolation formula for the spectra of power-law and log potentials. J. Phys. G Nucl. Part. Phys. 26, 981 (2000).","journal-title":"J. Phys. G Nucl. Part. Phys."},{"key":"16597_CR44","unstructured":"Herrmann, R. Solutions of the fractional Schr\u00f6dinger equation via diagonalization\u2014A plea for the harmonic oscillator basis. Part 1: The one dimensional case. arXiv:1805.03019 (2018)."},{"key":"16597_CR45","doi-asserted-by":"publisher","first-page":"052110","DOI":"10.1103\/PhysRevE.93.052110","volume":"93","author":"EV Kirichenko","year":"2016","unstructured":"Kirichenko, E. V., Garbaczewski, P., Stephanovich, V. & \u017baba, M. L\u00e9vy flights in an infinite potential well as a hypersingular Fredholm problem. Phys. Rev. E 93, 052110 (2016).","journal-title":"Phys. Rev. E"}],"container-title":["Scientific Reports"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.nature.com\/articles\/s41598-022-16597-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.nature.com\/articles\/s41598-022-16597-2","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.nature.com\/articles\/s41598-022-16597-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,11,24]],"date-time":"2023-11-24T21:37:16Z","timestamp":1700861836000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.nature.com\/articles\/s41598-022-16597-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,22]]},"references-count":45,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2022,12]]}},"alternative-id":["16597"],"URL":"https:\/\/doi.org\/10.1038\/s41598-022-16597-2","relation":{},"ISSN":["2045-2322"],"issn-type":[{"value":"2045-2322","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,7,22]]},"assertion":[{"value":"23 February 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 July 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 July 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no competing interests.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"12540"}}