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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-\\"},"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":"1704.05728","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2017-04-19T13:39:57Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"5621db5a6149e226d4f1e150bfb90564c3108a51a50682b8dbeffe62e09a1ae3","abstract_canon_sha256":"9c81706ea8e330e03b8b7e4daef72746c5bd8f9155a907919e1cd0d1403255c2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:46:06.678306Z","signature_b64":"dynJKtjQzIO2Ch8/mWJY3ATGcc5NL54GG2yQHBAqbIgcCXtLXE/j7ubaouHq4Iy79zcmidnu2Ij0aVxaP92RBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"109f0f6229299f0f4a03a8a0c1975b5279daf3fb571241964a3bf0d9a1fe0b15","last_reissued_at":"2026-05-18T00:46:06.677933Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:46:06.677933Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Parseval frames of exponentially decaying composite Wannier functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"David Auckly, Peter Kuchment","submitted_at":"2017-04-19T13:39:57Z","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. 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