{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,15]],"date-time":"2025-11-15T05:29:05Z","timestamp":1763184545490,"version":"3.45.0"},"reference-count":25,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2025,11,13]],"date-time":"2025-11-13T00:00:00Z","timestamp":1762992000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100006407","name":"Natural Science Foundation of Henan Province","doi-asserted-by":"publisher","award":["242300420239"],"award-info":[{"award-number":["242300420239"]}],"id":[{"id":"10.13039\/501100006407","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12401128"],"award-info":[{"award-number":["12401128"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Key Scientific Research Projects of Higher Education Institutions in Henan Province","award":["25B110002"],"award-info":[{"award-number":["25B110002"]}]},{"DOI":"10.13039\/501100018542","name":"Natural Science Foundation of Sichuan Province","doi-asserted-by":"publisher","award":["2024NSFSC1343"],"award-info":[{"award-number":["2024NSFSC1343"]}],"id":[{"id":"10.13039\/501100018542","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100017630","name":"Humanities and Social Sciences Youth Foundation of Ministry of Education of China","doi-asserted-by":"crossref","award":["24YJCZH108"],"award-info":[{"award-number":["24YJCZH108"]}],"id":[{"id":"10.13039\/501100017630","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Here we consider the problem of the Schr\u00f6dinger equation coupled with a neutral scalar field. 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