{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:GW7TN2QYAMCKU7XGDMD3NKYSTB","short_pith_number":"pith:GW7TN2QY","canonical_record":{"source":{"id":"1906.02186","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2019-06-05T11:06:43Z","cross_cats_sorted":[],"title_canon_sha256":"a749fc5883105744d73b7418abb68d879e620f0ffd89ed97931bbba31bf8ba7b","abstract_canon_sha256":"2063a5d458961b91104ae9d11f09c5e59b1649415aa6ff205fd86ba3a294cb0a"},"schema_version":"1.0"},"canonical_sha256":"35bf36ea180304aa7ee61b07b6ab129852be59a0f75f09beeae3a6e23a418b43","source":{"kind":"arxiv","id":"1906.02186","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1906.02186","created_at":"2026-05-17T23:44:01Z"},{"alias_kind":"arxiv_version","alias_value":"1906.02186v1","created_at":"2026-05-17T23:44:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1906.02186","created_at":"2026-05-17T23:44:01Z"},{"alias_kind":"pith_short_12","alias_value":"GW7TN2QYAMCK","created_at":"2026-05-18T12:33:18Z"},{"alias_kind":"pith_short_16","alias_value":"GW7TN2QYAMCKU7XG","created_at":"2026-05-18T12:33:18Z"},{"alias_kind":"pith_short_8","alias_value":"GW7TN2QY","created_at":"2026-05-18T12:33:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:GW7TN2QYAMCKU7XGDMD3NKYSTB","target":"record","payload":{"canonical_record":{"source":{"id":"1906.02186","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2019-06-05T11:06:43Z","cross_cats_sorted":[],"title_canon_sha256":"a749fc5883105744d73b7418abb68d879e620f0ffd89ed97931bbba31bf8ba7b","abstract_canon_sha256":"2063a5d458961b91104ae9d11f09c5e59b1649415aa6ff205fd86ba3a294cb0a"},"schema_version":"1.0"},"canonical_sha256":"35bf36ea180304aa7ee61b07b6ab129852be59a0f75f09beeae3a6e23a418b43","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:44:01.935114Z","signature_b64":"4vl98xkXWFr3SowHhWqHG0sJ1DYilZa2iQyp3q8gjbNIeDNOkxIoZQpgWaAruaLNcXOWqEg3l7dqn8cmTeTXDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"35bf36ea180304aa7ee61b07b6ab129852be59a0f75f09beeae3a6e23a418b43","last_reissued_at":"2026-05-17T23:44:01.934599Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:44:01.934599Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1906.02186","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:44:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"a5iDOKHQ6KH+Ji3QkHA2e95Sg7p/ksaFFQBKXo/AzMqRbdb7E1iCXiwzOdMb6Lrt6+Ci0yC84osAspXkRbklBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T10:20:18.585195Z"},"content_sha256":"86ab8cf2bd6ead7aacd6229ea77faecd1f5542084b9a7008efb158a21836a966","schema_version":"1.0","event_id":"sha256:86ab8cf2bd6ead7aacd6229ea77faecd1f5542084b9a7008efb158a21836a966"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:GW7TN2QYAMCKU7XGDMD3NKYSTB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Conditions for discreteness of the spectrum to multi-dimensional Schr\\\"odinger operator","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"Leonid Zelenko","submitted_at":"2019-06-05T11:06:43Z","abstract_excerpt":"This work is a continuation of our previos paper \\cite{Zel1}, where for the the Schr\\\"odinger operator $H=-\\Delta+ V(\\e)\\cdot$ $(V(\\e)\\ge 0)$, acting in the space $L_2(\\R^d)\\,(d\\ge 3)$, some constructive sufficient conditions for discreteness of its spectrum have been obtained on the base of well known Mazya -Shubin criterion and an optimization problem for a set function. Using a {\\it capacitary strong type inequality} of David Adams, the concept of {\\it base polyhedron} for the harmonic capacity and some properties of Choquet integral by this capacity, we obtain more general sufficient condi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.02186","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:44:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BHmkfx9zhuS5H7C4O9ndywMqJ2mgEPlESRiCR85ahaDhchmMHfMBR2NaAQw7Vvl4lKzShlaNnggf87z5jkPoBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T10:20:18.585614Z"},"content_sha256":"28f4ddd4434100d90d825dae56ff3f92faab2182e7d3e8cccfcae1e62fe189a1","schema_version":"1.0","event_id":"sha256:28f4ddd4434100d90d825dae56ff3f92faab2182e7d3e8cccfcae1e62fe189a1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GW7TN2QYAMCKU7XGDMD3NKYSTB/bundle.json","state_url":"https://pith.science/pith/GW7TN2QYAMCKU7XGDMD3NKYSTB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GW7TN2QYAMCKU7XGDMD3NKYSTB/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-31T10:20:18Z","links":{"resolver":"https://pith.science/pith/GW7TN2QYAMCKU7XGDMD3NKYSTB","bundle":"https://pith.science/pith/GW7TN2QYAMCKU7XGDMD3NKYSTB/bundle.json","state":"https://pith.science/pith/GW7TN2QYAMCKU7XGDMD3NKYSTB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GW7TN2QYAMCKU7XGDMD3NKYSTB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:GW7TN2QYAMCKU7XGDMD3NKYSTB","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":"2063a5d458961b91104ae9d11f09c5e59b1649415aa6ff205fd86ba3a294cb0a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2019-06-05T11:06:43Z","title_canon_sha256":"a749fc5883105744d73b7418abb68d879e620f0ffd89ed97931bbba31bf8ba7b"},"schema_version":"1.0","source":{"id":"1906.02186","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1906.02186","created_at":"2026-05-17T23:44:01Z"},{"alias_kind":"arxiv_version","alias_value":"1906.02186v1","created_at":"2026-05-17T23:44:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1906.02186","created_at":"2026-05-17T23:44:01Z"},{"alias_kind":"pith_short_12","alias_value":"GW7TN2QYAMCK","created_at":"2026-05-18T12:33:18Z"},{"alias_kind":"pith_short_16","alias_value":"GW7TN2QYAMCKU7XG","created_at":"2026-05-18T12:33:18Z"},{"alias_kind":"pith_short_8","alias_value":"GW7TN2QY","created_at":"2026-05-18T12:33:18Z"}],"graph_snapshots":[{"event_id":"sha256:28f4ddd4434100d90d825dae56ff3f92faab2182e7d3e8cccfcae1e62fe189a1","target":"graph","created_at":"2026-05-17T23:44:01Z","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":"This work is a continuation of our previos paper \\cite{Zel1}, where for the the Schr\\\"odinger operator $H=-\\Delta+ V(\\e)\\cdot$ $(V(\\e)\\ge 0)$, acting in the space $L_2(\\R^d)\\,(d\\ge 3)$, some constructive sufficient conditions for discreteness of its spectrum have been obtained on the base of well known Mazya -Shubin criterion and an optimization problem for a set function. Using a {\\it capacitary strong type inequality} of David Adams, the concept of {\\it base polyhedron} for the harmonic capacity and some properties of Choquet integral by this capacity, we obtain more general sufficient condi","authors_text":"Leonid Zelenko","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2019-06-05T11:06:43Z","title":"Conditions for discreteness of the spectrum to multi-dimensional Schr\\\"odinger operator"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.02186","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":"sha256:86ab8cf2bd6ead7aacd6229ea77faecd1f5542084b9a7008efb158a21836a966","target":"record","created_at":"2026-05-17T23:44:01Z","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":"2063a5d458961b91104ae9d11f09c5e59b1649415aa6ff205fd86ba3a294cb0a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2019-06-05T11:06:43Z","title_canon_sha256":"a749fc5883105744d73b7418abb68d879e620f0ffd89ed97931bbba31bf8ba7b"},"schema_version":"1.0","source":{"id":"1906.02186","kind":"arxiv","version":1}},"canonical_sha256":"35bf36ea180304aa7ee61b07b6ab129852be59a0f75f09beeae3a6e23a418b43","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"35bf36ea180304aa7ee61b07b6ab129852be59a0f75f09beeae3a6e23a418b43","first_computed_at":"2026-05-17T23:44:01.934599Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:44:01.934599Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4vl98xkXWFr3SowHhWqHG0sJ1DYilZa2iQyp3q8gjbNIeDNOkxIoZQpgWaAruaLNcXOWqEg3l7dqn8cmTeTXDg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:44:01.935114Z","signed_message":"canonical_sha256_bytes"},"source_id":"1906.02186","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:86ab8cf2bd6ead7aacd6229ea77faecd1f5542084b9a7008efb158a21836a966","sha256:28f4ddd4434100d90d825dae56ff3f92faab2182e7d3e8cccfcae1e62fe189a1"],"state_sha256":"357e41edc71135e392e725044ac17836c821ceb5c24b2cac6405c80a12dd0594"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Y5Q665K5AWNm1c1CwVGGWsbr4QV30f/ALbPoaz4ojM09Gi0kbq4fOvjZT/ni7SO6Zd9aAQC0ogbn2+wi6zD8Bg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T10:20:18.588213Z","bundle_sha256":"8f43fc2bdd3a3be028876a3aee304661ca97f319ae90eb1dcdb2f43a311f702c"}}