{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T23:55:55Z","timestamp":1767916555566,"version":"3.49.0"},"reference-count":52,"publisher":"Frontiers Media SA","license":[{"start":{"date-parts":[[2025,7,10]],"date-time":"2025-07-10T00:00:00Z","timestamp":1752105600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["frontiersin.org"],"crossmark-restriction":true},"short-container-title":["Front. Appl. Math. Stat."],"abstract":"<jats:p>In this paper, we investigate negative eigenvalues of exactly solvable quantum models, particularly one-dimensional Hamiltonians with \u03b4\u2032-like potentials used to represent localized dipoles. These operators arise as norm resolvent limits of Schr\u00f6dinger operators with suitably regularized potentials. Although the limiting operator is bounded below, we show that the approximating operators may possess a finite but arbitrarily large number of negative eigenvalues that diverge to \u2212\u221e as the regularization parameter vanishes. This phenomenon illustrates a spectral instability of Schr\u00f6dinger operators with \u03b4\u2032-like singularities.<\/jats:p>","DOI":"10.3389\/fams.2025.1615447","type":"journal-article","created":{"date-parts":[[2025,7,10]],"date-time":"2025-07-10T05:31:46Z","timestamp":1752125506000},"update-policy":"https:\/\/doi.org\/10.3389\/crossmark-policy","source":"Crossref","is-referenced-by-count":2,"title":["On negative eigenvalues of 1D Schr\u00f6dinger operators with \u03b4\u2032-like potentials"],"prefix":"10.3389","volume":"11","author":[{"given":"Yuriy","family":"Golovaty","sequence":"first","affiliation":[]},{"given":"Rostyslav","family":"Hryniv","sequence":"additional","affiliation":[]}],"member":"1965","published-online":{"date-parts":[[2025,7,10]]},"reference":[{"key":"B1","volume-title":"Solvable Models in Quantum Mechanics","author":"Albeverio","year":"2005"},{"key":"B2","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511758904","author":"Albeverio","year":"2000","journal-title":"Singular Perturbations of Differential Operators. 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