{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:41:37Z","timestamp":1760146897944,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2024,12,19]],"date-time":"2024-12-19T00:00:00Z","timestamp":1734566400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["AppliedMath"],"abstract":"<jats:p>We study a separable Hilbert space of smooth curves taking values in the Segal\u2013Bergmann space of analytic functions in the complex plane, and two of its subspaces that are the domains of unbounded non self-adjoint linear partial differential operators of the first and second order. We show how to build a Hilbert basis for this space. We study these first- and second-order partial derivation non-self-adjoint operators defined on this space, showing that these operators are defined on dense subspaces of the initial space of smooth curves; we determine their respective adjoints, compute their respective commutators, determine their eigenvalues and, under some normalisation conditions on the eigenvectors, we present examples of a discrete set of eigenvalues. Using these derivation operators, we study a Schr\u00f6dinger-type equation, building particular solutions given by their representation as smooth curves on the Segal\u2013Bergmann space, and we show the existence of general solutions using an Fourier\u2013Hilbert base of the space of smooth curves. We point out the existence of self-adjoint operators in the space of smooth curves that are obtained by the composition of the partial derivation operators with multiplication operators, showing that these operators admit simple sequences of eigenvalues and eigenvectors. We present two applications of the Schr\u00f6dinger-type equation studied. In the first one, we consider a wave associated with an object having the mass of an electron, showing that two waves, when considered as having only a free real space variable, are entangled, in the sense that the probability densities in the real variable are almost perfectly correlated. In the second application, after postulating that a usual package of information may have a mass of the order of magnitude of the neutron\u2019s mass attributed to it\u2014and so well into the domain of possible quantisation\u2014we explore some consequences of the model.<\/jats:p>","DOI":"10.3390\/appliedmath4040083","type":"journal-article","created":{"date-parts":[[2024,12,19]],"date-time":"2024-12-19T10:54:20Z","timestamp":1734605660000},"page":"1555-1587","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On a Schr\u00f6dinger Equation in the Complex Space Variable"],"prefix":"10.3390","volume":"4","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4991-7568","authenticated-orcid":false,"given":"Manuel L.","family":"Esqu\u00edvel","sequence":"first","affiliation":[{"name":"Department of Mathematics, Nova School of Science and Technology and Nova Math, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4226-1658","authenticated-orcid":false,"given":"Nadezhda P.","family":"Krasii","sequence":"additional","affiliation":[{"name":"Nova Math, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal"},{"name":"Department of Higher Mathematics, Don State Technical University, Gagarin Square 1, 344000 Rostov-on-Don, Russia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4963-3592","authenticated-orcid":false,"given":"Philippe L.","family":"Didier","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Nova School of Science and Technology and Nova Math, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2024,12,19]]},"reference":[{"key":"ref_1","first-page":"371","article-title":"On the nonlinear Schr\u00f6dinger equation in spaces of infinite mass and low regularity","volume":"35","author":"Barros","year":"2022","journal-title":"Differ. Integral Equ."},{"key":"ref_2","unstructured":"Thom, R. (1980). Mod\u00e8les Math\u00e9matiques de la Morphog\u00e9n\u00e8se, Christian Bourgeois \u00c9diteur. [2nd ed.]."},{"key":"ref_3","unstructured":"Thom, R. (1977). Stabilit\u00e9 Structurelle et Morphog\u00e9n\u00e8se, Inter Editions. [2nd ed.]. Essai d\u2019une th\u00e9orie g\u00e9n\u00e9rale des mod\u00e8les."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Berezin, F.A., and Shubin, M.A. (1991). The Schr\u00f6dinger Equation, Kluwer Academic Publishers Group. [Russian ed.].","DOI":"10.1007\/978-94-011-3154-4"},{"key":"ref_5","first-page":"445","article-title":"Some Hilbert spaces of analytic functions. I","volume":"106","year":"1963","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_6","unstructured":"de Branges, L. (1968). Hilbert Spaces of Entire Functions, Prentice-Hall, Inc."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"653","DOI":"10.1016\/0022-247X(77)90252-9","article-title":"Schr\u00f6dinger-Dirac spaces of entire functions","volume":"57","year":"1977","journal-title":"J. Math. Anal. Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1512\/iumj.1982.31.31017","article-title":"The operator Mz on Hilbert spaces of analytic functions","volume":"31","author":"Shelburne","year":"1982","journal-title":"Indiana Univ. Math. J."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"49","DOI":"10.4064\/sm-108-1-49-54","article-title":"Operators on spaces of analytic functions","volume":"108","author":"Seddighi","year":"1994","journal-title":"Studia Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1019","DOI":"10.1016\/j.jmaa.2017.05.021","article-title":"Composition operators on Hilbert spaces of entire functions with analytic symbols","volume":"454","author":"Stochel","year":"2017","journal-title":"J. Math. Anal. Appl."},{"key":"ref_11","unstructured":"Blank, J., Exner, P., and Havl\u00ed\u010dek, M. (2008). Hilbert Space Operators in Quantum Physics, AIP Press. [2nd ed.]. Theoretical and Mathematical Physics."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"262","DOI":"10.3844\/jmssp.2014.262.266","article-title":"Stationary Connected Curves in Hilbert Spaces","volume":"10","author":"Hatamleh","year":"2014","journal-title":"J. Math. Stat."},{"key":"ref_13","unstructured":"Schul, R. (2005). Subset of Rectifiable Curves in Hilbert Space and the Analyst\u2019s TSP. [Ph.D. Thesis, Yale University]."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"659","DOI":"10.3390\/quantum5040042","article-title":"A Schr\u00f6dinger Equation for Evolutionary Dynamics","volume":"5","author":"Ao","year":"2023","journal-title":"Quantum Rep."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Hu, J., and Zhang, N. (2024). A \u2202-Dressing Method for the Kundu-Nonlinear Schr\u00f6dinger Equation. Mathematics, 12.","DOI":"10.3390\/math12020278"},{"key":"ref_16","unstructured":"Neveu, J. (1968). Processus Al\u00e9atoires Gaussiens, Les Presses de l\u2019Universit\u00e9 de Montr\u00e9al. S\u00e9minaire de Math\u00e9matiques Sup\u00e9rieures [Seminar on Higher Mathematics]."},{"key":"ref_17","unstructured":"Lang, S. (1983). Real Analysis, Addison-Wesley Publishing Company, Advanced Book Program. [2nd ed.]."},{"key":"ref_18","unstructured":"Katznelson, Y. (1976). An Introduction to Harmonic Analysis, Dover Publications, Inc.. [Corrected ed.]."},{"key":"ref_19","unstructured":"Young, R.M. (2001). An Introduction to Nonharmonic Fourier Series, Academic Press, Inc.. [1st ed.]."},{"key":"ref_20","unstructured":"Rudin, W. (1987). Real and Complex Analysis, McGraw-Hill Book Co.. [3rd ed.]."},{"key":"ref_21","unstructured":"Rudin, W. (1991). Functional Analysis, McGraw-Hill, Inc.. [2nd ed.]."},{"key":"ref_22","unstructured":"Silverman, R.A., and Friedman, M.D. (1957). Mathematical Foundations of Information Theory, Dover Publications, Inc."},{"key":"ref_23","unstructured":"Fagas, G., Gammaitoni, L., Paul, D., and Berini, G.A. (2014). Minimum Energy of Computing, Fundamental Considerations. ICT-Energy-Concepts Towards Zero, IntechOpen. Chapter 7."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"905","DOI":"10.1007\/BF02084158","article-title":"The thermodynamics of computation\u2014A review","volume":"21","author":"Bennett","year":"1982","journal-title":"Int. J. Theor. Phys."},{"key":"ref_25","unstructured":"Burks, A.W. (1966). Theory of Self-Reproducing Automata, University of Illinois Press."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1103\/PhysRevD.23.287","article-title":"Universal upper bound on the entropy-to-energy ratio for bounded systems","volume":"23","author":"Bekenstein","year":"1981","journal-title":"Phys. Rev. D"}],"container-title":["AppliedMath"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2673-9909\/4\/4\/83\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:55:43Z","timestamp":1760115343000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2673-9909\/4\/4\/83"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,12,19]]},"references-count":26,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2024,12]]}},"alternative-id":["appliedmath4040083"],"URL":"https:\/\/doi.org\/10.3390\/appliedmath4040083","relation":{},"ISSN":["2673-9909"],"issn-type":[{"type":"electronic","value":"2673-9909"}],"subject":[],"published":{"date-parts":[[2024,12,19]]}}}