{"paper":{"title":"A Brunn-Minkowski inequality for Schr\\\"odinger operators with Kato class potentials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Schrödinger operators with convex Kato potentials obey a Brunn-Minkowski inequality on their first Dirichlet eigenvalue.","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alessandro Carbotti","submitted_at":"2026-03-31T16:54:27Z","abstract_excerpt":"In this paper we prove a Brunn-Minkowski inequality for the first Dirichlet eigenvalue of a Schr\\\"odinger type operator $\\mathcal{H}_V:=-\\operatorname{div}(A\\nabla)+V$, where $V$ is convex and Kato decomposable, using the trace class property of the generated semigroup. As a consequence, we obtain the log-concavity of the ground state using the ultracontractivity of the semigroup, and also the strong log-concavity under additional assumptions on $\\Omega$ and $V$."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"In this paper we prove a Brunn-Minkowski inequality for the first Dirichlet eigenvalue of a Schrödinger type operator H_V := -div(A ∇) + V, where V is convex and Kato decomposable, using the trace class property of the generated semigroup.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The potential V must be convex and Kato decomposable, and the semigroup generated by the operator must be trace class (and ultracontractive for the log-concavity conclusions).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A Brunn-Minkowski inequality holds for the ground-state eigenvalue of Schrödinger operators with convex Kato decomposable potentials.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Schrödinger operators with convex Kato potentials obey a Brunn-Minkowski inequality on their first Dirichlet eigenvalue.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"017b324f9f48b91c1fe31bda2733766c16e8027c6e50eea8b064e62e642f8fee"},"source":{"id":"2603.29989","kind":"arxiv","version":5},"verdict":{"id":"53c9966a-326e-4095-9419-a77ceb7b3b2e","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-13T22:55:58.307365Z","strongest_claim":"In this paper we prove a Brunn-Minkowski inequality for the first Dirichlet eigenvalue of a Schrödinger type operator H_V := -div(A ∇) + V, where V is convex and Kato decomposable, using the trace class property of the generated semigroup.","one_line_summary":"A Brunn-Minkowski inequality holds for the ground-state eigenvalue of Schrödinger operators with convex Kato decomposable potentials.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The potential V must be convex and Kato decomposable, and the semigroup generated by the operator must be trace class (and ultracontractive for the log-concavity conclusions).","pith_extraction_headline":"Schrödinger operators with convex Kato potentials obey a Brunn-Minkowski inequality on their first Dirichlet eigenvalue."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2603.29989/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"c6487a24f1fe0e70ab8ae7403dedf93d05ca255765d405d52c2fd38b96a537cb"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}