{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,14]],"date-time":"2025-10-14T00:48:42Z","timestamp":1760402922265,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2021,4,9]],"date-time":"2021-04-09T00:00:00Z","timestamp":1617926400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In applications, many states given for a system can be expressed by orthonormal elements, called \u201cstate elements\u201d, taken in a separable Hilbert space (called \u201cstate space\u201d). The exact nature of the Hilbert space depends on the system; for example, the state space for position and momentum states is the space of square-integrable functions. The symmetries of a quantum system can be represented by a class of unitary operators that act in the Hilbert space. The operators called ladder operators have the effect of lowering or raising the energy of the state. In this paper, we study the spectral properties of a self-adjoint, fourth-order differential operator with a bounded operator coefficient and establish a second regularized trace formula for this operator.<\/jats:p>","DOI":"10.3390\/sym13040629","type":"journal-article","created":{"date-parts":[[2021,4,12]],"date-time":"2021-04-12T11:05:06Z","timestamp":1618225506000},"page":"629","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["A Second Regularized Trace Formula for a Fourth Order Differential Operator"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0626-0148","authenticated-orcid":false,"given":"Erdal","family":"G\u00fcl","sequence":"first","affiliation":[{"name":"Department of Mathematics, Yildiz Technical University, Esenler, 34220 Istanbul, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Aylan","family":"Ceyhan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Yildiz Technical University, Esenler, 34220 Istanbul, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,4,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"715","DOI":"10.1515\/JIIP.2008.044","article-title":"Inverse nodal problems for Sturm-Liouville operators on star-type graphs","volume":"16","author":"Yurko","year":"2008","journal-title":"J. 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