{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:CCPQ6YRJFGPQ6SQDVCQMDF23KJ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"9c81706ea8e330e03b8b7e4daef72746c5bd8f9155a907919e1cd0d1403255c2","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2017-04-19T13:39:57Z","title_canon_sha256":"5621db5a6149e226d4f1e150bfb90564c3108a51a50682b8dbeffe62e09a1ae3"},"schema_version":"1.0","source":{"id":"1704.05728","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1704.05728","created_at":"2026-05-18T00:46:06Z"},{"alias_kind":"arxiv_version","alias_value":"1704.05728v1","created_at":"2026-05-18T00:46:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1704.05728","created_at":"2026-05-18T00:46:06Z"},{"alias_kind":"pith_short_12","alias_value":"CCPQ6YRJFGPQ","created_at":"2026-05-18T12:31:10Z"},{"alias_kind":"pith_short_16","alias_value":"CCPQ6YRJFGPQ6SQD","created_at":"2026-05-18T12:31:10Z"},{"alias_kind":"pith_short_8","alias_value":"CCPQ6YRJ","created_at":"2026-05-18T12:31:10Z"}],"graph_snapshots":[{"event_id":"sha256:515163c4a0eec1db3a6340da813eb6be82c2e68264ee8d950ba1a4b00c0d4dcb","target":"graph","created_at":"2026-05-18T00:46:06Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $L$ be a periodic self-adjoint linear elliptic operator in $\\R^n$ with coefficients periodic with respect to a lattice $\\G$, e.g. Schr\\\"{o}dinger operator $(i^{-1}\\partial/\\partial_x-A(x))^2+V(x)$ with periodic magnetic and electric potentials $A,V$, or a Maxwell operator $\\nabla\\times\\varepsilon (x)^{-1}\\nabla\\times$ in a periodic medium. Let also $S$ be a finite part of its spectrum separated by gaps from the rest of the spectrum. We address here the question of existence of a finite set of exponentially decaying Wannier functions $w_j(x)$ such that their $\\G$-shifts $w_{j,\\g}(x)=w_j(x-\\","authors_text":"David Auckly, Peter Kuchment","cross_cats":["math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2017-04-19T13:39:57Z","title":"On Parseval frames of exponentially decaying composite Wannier functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.05728","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:6ecbf98efb9750f607038bff0f3d33dfcf0f44339cb938261c25f4eafd9dd28f","target":"record","created_at":"2026-05-18T00:46:06Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"9c81706ea8e330e03b8b7e4daef72746c5bd8f9155a907919e1cd0d1403255c2","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2017-04-19T13:39:57Z","title_canon_sha256":"5621db5a6149e226d4f1e150bfb90564c3108a51a50682b8dbeffe62e09a1ae3"},"schema_version":"1.0","source":{"id":"1704.05728","kind":"arxiv","version":1}},"canonical_sha256":"109f0f6229299f0f4a03a8a0c1975b5279daf3fb571241964a3bf0d9a1fe0b15","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"109f0f6229299f0f4a03a8a0c1975b5279daf3fb571241964a3bf0d9a1fe0b15","first_computed_at":"2026-05-18T00:46:06.677933Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:46:06.677933Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"dynJKtjQzIO2Ch8/mWJY3ATGcc5NL54GG2yQHBAqbIgcCXtLXE/j7ubaouHq4Iy79zcmidnu2Ij0aVxaP92RBg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:46:06.678306Z","signed_message":"canonical_sha256_bytes"},"source_id":"1704.05728","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6ecbf98efb9750f607038bff0f3d33dfcf0f44339cb938261c25f4eafd9dd28f","sha256:515163c4a0eec1db3a6340da813eb6be82c2e68264ee8d950ba1a4b00c0d4dcb"],"state_sha256":"9dc6f87df1d23021228cab835cfa11d03f6cee7d0016d14227bd66e7f87f35eb"}