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For instance, let $L$ be the generator of a L\\'evy process with L\\'evy measure $\\nu(d z):= \\rho(z) d z$ such that $\\rho(z)=\\rho(-z)$ and $$c_1 |z|^{-(d+\\alpha_1)}\\le \\rho(z)\\le c_2|z|^{-(d+\\alpha_2)},\\ \\ |z|\\le \\kappa $$ for some constants $\\kappa, c_1,c_2>0$ and $\\alpha_1,\\alpha_2\\in (0,2),$ and let $c_3|x|^{\\theta_1} \\le V(x)\\le c_4|x|^{\\theta_2}$ for some constants $\\theta_1,\\theta_2, c_3,c_4>0$ and large $|x|$. Then the eige"},"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":"1703.09954","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2017-03-29T09:52:57Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"7e02e0ce8bfe39df2a7b54f319a8589f5edfd51816b65ad3b3b73e8cf110a521","abstract_canon_sha256":"0cf883f2eefaad48b842833d053048d42d4e9b0425cea38a0cb22f202c381e78"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:40:52.604082Z","signature_b64":"kMzwv8Ok2ixXpFYgxl0oiFM/NR25x+pf9+TUkHiKZrZneYVlZHMqoAsjIgV1k6xEvN1hlfL+yvvvJb+tGyLEDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c369e44377fa30932b11f19acd1769b3a6ac49e2a1b11c7104904e09f50086b9","last_reissued_at":"2026-05-18T00:40:52.603474Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:40:52.603474Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Higher Order Eigenvalues for Non-Local Schr\\\"odinger Operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Feng-Yu Wang, Niels Jacob","submitted_at":"2017-03-29T09:52:57Z","abstract_excerpt":"Two-sided estimates for higher order eigenvalues are presented for a class of non-local Schr\\\"odinger operators by using the jump rate and the growth of the potential. For instance, let $L$ be the generator of a L\\'evy process with L\\'evy measure $\\nu(d z):= \\rho(z) d z$ such that $\\rho(z)=\\rho(-z)$ and $$c_1 |z|^{-(d+\\alpha_1)}\\le \\rho(z)\\le c_2|z|^{-(d+\\alpha_2)},\\ \\ |z|\\le \\kappa $$ for some constants $\\kappa, c_1,c_2>0$ and $\\alpha_1,\\alpha_2\\in (0,2),$ and let $c_3|x|^{\\theta_1} \\le V(x)\\le c_4|x|^{\\theta_2}$ for some constants $\\theta_1,\\theta_2, c_3,c_4>0$ and large $|x|$. 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