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We are then interested in the spectrum of $H_t = -\\Delta + V_t$, for $t \\in \\mathbb R$, where $V_t(x,y) = V^{(1)}(x,y)$, for $x > 0$, and $V_t(x,y) = V^{(2)}(x+t,y)$, for $x < 0$. While the essential spectrum of $H_t$ is independent of $t$, we show that discrete spectrum, related to the interface at"},"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":"1412.6420","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2014-12-19T16:31:11Z","cross_cats_sorted":["math.MP","math.SP"],"title_canon_sha256":"80c35296a809af86107c4ac67dfdd0de5e3c7f34064ed8232eb27465e5805148","abstract_canon_sha256":"761ff9d7a7615db535a6c30db4664e2804385814731222e1a586f07af1a426cc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:35:35.131648Z","signature_b64":"8zOO5aPjylUMl7/84PYU/qMtjvblkdsYtVVsUij8hZ0RGD4B1eDBalTWiP+lrz7863ZuMpaaPMnlPSWVWV+iCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a7b1068a2e300fa81c07487dbf2bbe5577b7aa013e53f0c45b50b71125163a1c","last_reissued_at":"2026-05-18T01:35:35.131035Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:35:35.131035Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Bound States for Nano-Tubes with a Dislocation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP","math.SP"],"primary_cat":"math-ph","authors_text":"J\\\"urgen Voigt, Marko Stautz, Martin Kohlmann, Rainer Hempel","submitted_at":"2014-12-19T16:31:11Z","abstract_excerpt":"As a model for an interface in solid state physics, we consider two real-valued potentials $V^{(1)}$ and $V^{(2)}$ on the cylinder or tube $S=\\mathbb R \\times (\\mathbb R/\\mathbb Z)$ where we assume that there exists an interval $(a_0,b_0)$ which is free of spectrum of $-\\Delta+V^{(k)}$ for $k=1,2$. We are then interested in the spectrum of $H_t = -\\Delta + V_t$, for $t \\in \\mathbb R$, where $V_t(x,y) = V^{(1)}(x,y)$, for $x > 0$, and $V_t(x,y) = V^{(2)}(x+t,y)$, for $x < 0$. 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