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For the zero energy eigenvalue the equations for all $a$ are reduced to the same equation representing two-dimensional free motions in the constant potential $V_a=-g_a$ in terms of the conformal mappings of $\\zeta_a=z^a$ with $z=x+iy$. Namely, the zero energy eigenstates are described by the plane waves with the fixed wave numbers $k_a=\\sqrt{mg_a}/\\hb"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"cond-mat/0103209","kind":"arxiv","version":4},"metadata":{"license":"","primary_cat":"cond-mat.mes-hall","submitted_at":"2001-03-09T06:20:57Z","cross_cats_sorted":["cond-mat.mtrl-sci"],"title_canon_sha256":"ca24edb5518cc04a65ab0a2d94d94d7cd7d10c0c0cb6010ca419e0440bb55e4b","abstract_canon_sha256":"af3359b2dcb5ed44e34f60c62ab3ac0b4f4624fdcfd3e12f89416332188dfd14"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:07:03.338412Z","signature_b64":"lKIohT67FjAS6TjTwS411Y4/EUg9ydA+iJf6HjIldHdzxNnKaJdqvaHznrmFHWAm5x/eiWryheeBeJT/R5TnAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a6423f828b4f09393838eecd31b8a1f31067e1583203054857df56868aa5613f","last_reissued_at":"2026-05-18T01:07:03.337795Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:07:03.337795Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Zero Energy Solutions and Vortices in Schroedinger Equations","license":"","headline":"","cross_cats":["cond-mat.mtrl-sci"],"primary_cat":"cond-mat.mes-hall","authors_text":"Toshiki Shimbori, Tsunehiro Kobayashi","submitted_at":"2001-03-09T06:20:57Z","abstract_excerpt":"All two-dimensional Schr\\\"{o}dinger equations with symmetric potentials \\break $(V_a(\\rho)=-a^2g_a \\rho ^{2(a-1)/2} {with} \\rho=\\sqrt{x^2+y^2} {and} a\\not=0)$ is shown to have zero energy states contained in conjugate spaces of Gel'fand triplets. For the zero energy eigenvalue the equations for all $a$ are reduced to the same equation representing two-dimensional free motions in the constant potential $V_a=-g_a$ in terms of the conformal mappings of $\\zeta_a=z^a$ with $z=x+iy$. 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