files / problems / simon-lieb-thirring / compute / record.json

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{
  "classical": {
    "formula": "2^{-d} pi^{-d/2} Gamma(gamma+1)/Gamma(gamma+1+d/2)",
    "Lcl_1_1": 0.21220659078919374,
    "Lcl_1_1_closed": "2/(3pi)",
    "Lcl_1_1_closed_value": 0.2122065907891938,
    "Lcl_3_2_1": 0.18750000000000003,
    "Lcl_3_2_1_closed": "3/16",
    "Lcl_3_2_1_closed_value": 0.1875,
    "Lcl_1_3": 0.0067547455761558526
  },
  "published": {
    "fhjn_theorem_1_ratio": 1.456,
    "fhjn_finer_ratio": 1.455786,
    "one_bound_state_over_classical_1d_gamma1": 1.1547005383792517,
    "citation": "Frank\u2013Hundertmark\u2013Jex\u2013Nam, arXiv:1808.09017 / JEMS 23 (2021)"
  },
  "sech2_witnesses": [
    {
      "nu": 1,
      "gamma": 1.0,
      "n_bound": 1,
      "moment": 1.0,
      "integral": 4.442882938158366,
      "ratio": 0.22507907903927651,
      "ratio_over_classical": 1.0606601717798214
    },
    {
      "nu": 2,
      "gamma": 1.0,
      "n_bound": 2,
      "moment": 5.0,
      "integral": 23.085896942913553,
      "ratio": 0.21658244478713226,
      "ratio_over_classical": 1.0206207261596578
    },
    {
      "nu": 3,
      "gamma": 1.0,
      "n_bound": 3,
      "moment": 14.0,
      "integral": 65.29677711243184,
      "ratio": 0.21440568155291914,
      "ratio_over_classical": 1.0103629710818454
    },
    {
      "nu": 4,
      "gamma": 1.0,
      "n_bound": 4,
      "moment": 30.0,
      "integral": 140.49629462081452,
      "ratio": 0.2135287630251531,
      "ratio_over_classical": 1.0062305898749055
    },
    {
      "nu": 1,
      "gamma": 1.25,
      "n_bound": 1,
      "moment": 1.0,
      "integral": 4.836056776129937,
      "ratio": 0.20678003718563695,
      "ratio_over_classical": 1.0405562256273813
    },
    {
      "nu": 2,
      "gamma": 1.25,
      "n_bound": 2,
      "moment": 6.656854249492381,
      "integral": 33.0714766470665,
      "ratio": 0.2012868769221666,
      "ratio_over_classical": 1.01291360505182
    },
    {
      "nu": 3,
      "gamma": 1.25,
      "n_bound": 3,
      "moment": 22.245311517612276,
      "integral": 111.23874463866059,
      "ratio": 0.1999780884787242,
      "ratio_over_classical": 1.0063275342618696
    }
  ],
  "note": "Witness ratios are lower bounds on L(gamma,1), not upper bounds. They do not improve 1.456."
}