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The criteria for $q$ such that $H=H_0+q$ has at most one eigenvalue (finitely many, infinitely many eigenvalues) are obtained.\n  All the results are quantitative and are generalized to the perturbed Stark type operator."},"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":"1809.04694","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2018-09-12T22:36:16Z","cross_cats_sorted":["math.MP","math.SP"],"title_canon_sha256":"48a97a2407899ffb3d3ed13cef47dfc6aa6fdc1e549af2c120c63bd59f9d12c7","abstract_canon_sha256":"fe2a18d15618f66991e578351adb728a1fa4be0ab04057f4217ea68017afcda4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:49:03.446680Z","signature_b64":"sfgLXmcgONwunwkpN7jO+RbSAtffIY0anThjW3g9j8RJAzS7oKVxZYz4LP0rklhu5RHxprQigEV0mpbsZcrEBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"18a6050a8a229893d8b5418d921de59d2530ed7830148fba502495aba27de670","last_reissued_at":"2026-05-17T23:49:03.446124Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:49:03.446124Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Criteria for eigenvalues embedded into the absolutely continuous spectrum of perturbed Stark type operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP","math.SP"],"primary_cat":"math-ph","authors_text":"Wencai Liu","submitted_at":"2018-09-12T22:36:16Z","abstract_excerpt":"In this paper, we consider the perturbed Stark operator \\begin{equation*}\n  Hu=H_0u+qu=-u^{\\prime\\prime}-xu+qu, \\end{equation*} where $q$ is the power-decaying perturbation. 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