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We study so called virtual bound levels of the operator $H_\\gamma$, that is those eigenvalues of $H_\\gamma$ which are born at the moment $\\gamma=0$ in a gap $(\\lambda_-,\\,\\lambda_+)$ of the spectrum of the unperturbed operator $H_0=-\\Delta+ V(x)\\cdot$ from an edge of"},"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":"1505.07381","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2015-05-27T15:39:35Z","cross_cats_sorted":[],"title_canon_sha256":"d65d8d1a5410cb5c1aed9dfacf0fd157b2b6c0133eb09373dbf79c32de9941a6","abstract_canon_sha256":"0558faa10e17cc713a7b634996d179f7f107bfc92405a3a94870e856d2c69d28"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:02:04.626408Z","signature_b64":"pV8WZLS0939GuSFLUzcdJ51dgOzcFt/nu5uQs0CdO0W/2PXg81ZboPfyJO7cM6klw79UeFtpQBMM3zxsddulBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6f9810a42d072fc6429cda71f95456bc93b4dca1399ae53832fca766660272da","last_reissued_at":"2026-05-18T02:02:04.625574Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:02:04.625574Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Virtual bound levels in a gap of the essential spectrum of the Schroedinger operator with a weakly perturbed periodic potential","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"Leonid Zelenko","submitted_at":"2015-05-27T15:39:35Z","abstract_excerpt":"In the space $L_2(R^d)$ we consider the Schr\\\"odinger operator $H_\\gamma=-\\Delta+ V(x)\\cdot+\\gamma W(x)\\cdot$, where $V(x)=V(x_1,x_2,\\dots,x_d)$ is a periodic function with respect to all the variables, $\\gamma$ is a small real coupling constant and the perturbation $W(x)$ tends to zero sufficiently fast as $|x|\\rightarrow\\infty$. We study so called virtual bound levels of the operator $H_\\gamma$, that is those eigenvalues of $H_\\gamma$ which are born at the moment $\\gamma=0$ in a gap $(\\lambda_-,\\,\\lambda_+)$ of the spectrum of the unperturbed operator $H_0=-\\Delta+ V(x)\\cdot$ from an edge of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.07381","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1505.07381","created_at":"2026-05-18T02:02:04.625723+00:00"},{"alias_kind":"arxiv_version","alias_value":"1505.07381v1","created_at":"2026-05-18T02:02:04.625723+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.07381","created_at":"2026-05-18T02:02:04.625723+00:00"},{"alias_kind":"pith_short_12","alias_value":"N6MBBJBNA4X4","created_at":"2026-05-18T12:29:32.376354+00:00"},{"alias_kind":"pith_short_16","alias_value":"N6MBBJBNA4X4MQU4","created_at":"2026-05-18T12:29:32.376354+00:00"},{"alias_kind":"pith_short_8","alias_value":"N6MBBJBN","created_at":"2026-05-18T12:29:32.376354+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/N6MBBJBNA4X4MQU43JY7SVCWXS","json":"https://pith.science/pith/N6MBBJBNA4X4MQU43JY7SVCWXS.json","graph_json":"https://pith.science/api/pith-number/N6MBBJBNA4X4MQU43JY7SVCWXS/graph.json","events_json":"https://pith.science/api/pith-number/N6MBBJBNA4X4MQU43JY7SVCWXS/events.json","paper":"https://pith.science/paper/N6MBBJBN"},"agent_actions":{"view_html":"https://pith.science/pith/N6MBBJBNA4X4MQU43JY7SVCWXS","download_json":"https://pith.science/pith/N6MBBJBNA4X4MQU43JY7SVCWXS.json","view_paper":"https://pith.science/paper/N6MBBJBN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1505.07381&json=true","fetch_graph":"https://pith.science/api/pith-number/N6MBBJBNA4X4MQU43JY7SVCWXS/graph.json","fetch_events":"https://pith.science/api/pith-number/N6MBBJBNA4X4MQU43JY7SVCWXS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/N6MBBJBNA4X4MQU43JY7SVCWXS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/N6MBBJBNA4X4MQU43JY7SVCWXS/action/storage_attestation","attest_author":"https://pith.science/pith/N6MBBJBNA4X4MQU43JY7SVCWXS/action/author_attestation","sign_citation":"https://pith.science/pith/N6MBBJBNA4X4MQU43JY7SVCWXS/action/citation_signature","submit_replication":"https://pith.science/pith/N6MBBJBNA4X4MQU43JY7SVCWXS/action/replication_record"}},"created_at":"2026-05-18T02:02:04.625723+00:00","updated_at":"2026-05-18T02:02:04.625723+00:00"}