range of r: -L..L with L=120 (open ends); convergence: comparison with L=240 and weight at |r|>L/2 finite differences: 5-point, step 0.01 in P, relative step 0.001 in M_i, tau_i, theta; convention eta=1/2 === set A_equal_R: {'M1': 1.0, 'M2': 1.0, 'tau1': 0.1, 'tau2': 0.1, 'w': [-0.004, 0.003, -0.002, 0.006]} max tau_i/M_i = 0.100 P=-0.30 E_b=1.998771748883 isolation=1.452e-03 in a band: False weight(r=0,11)=0.6408 Gamma=+1.000000 E_b'=-1.133090e-03 =-1.133090e-03 =-1.133090e-03 E_b''=+3.517649e-03 (11.9)=+3.517649e-03 P=-0.15 E_b=1.998642641792 isolation=1.414e-03 in a band: False weight(r=0,11)=0.6278 Gamma=+1.000000 E_b'=-5.810445e-04 =-5.810445e-04 =-5.810445e-04 E_b''=+3.809572e-03 (11.9)=+3.809572e-03 P=+0.00 E_b=1.998598883527 isolation=1.401e-03 in a band: False weight(r=0,11)=0.6235 Gamma=+1.000000 E_b'=+1.850372e-15 =+7.893878e-21 =-2.053590e-20 E_b''=+3.905596e-03 (11.9)=+3.905596e-03 P=+0.15 E_b=1.998642641792 isolation=1.414e-03 in a band: False weight(r=0,11)=0.6278 Gamma=+1.000000 E_b'=+5.810445e-04 =+5.810445e-04 =+5.810445e-04 E_b''=+3.809572e-03 (11.9)=+3.809572e-03 P=+0.30 E_b=1.998771748883 isolation=1.452e-03 in a band: False weight(r=0,11)=0.6408 Gamma=+1.000000 E_b'=+1.133090e-03 =+1.133090e-03 =+1.133090e-03 E_b''=+3.517649e-03 (11.9)=+3.517649e-03 max |deviation| over the P values: 2(a): dE/dM_i - 1.25e-12; tau_i dE/dtau_i - 1.37e-12; dE/dtheta - 8.96e-13; Euler residual 4.22e-15 2(b): E_b' - 1.59e-12; E_b' - 1.59e-12; - 6.51e-19 2(c): E_b'' (finite difference) - formula (11.9) 1.59e-11 3(a): coefficients of (11.10) against direct inversion (relative) 1.19e-14; root of (11.10) - E_b 1.55e-15 numerics: |E_b(L) - E_b(2L)| 2.00e-15; weight at |r|>L/2 3.17e-20; dependence on eta 2.44e-15; dense - banded 2.44e-15 closed forms (items 3, 4), regime R: |w11|/a = 0.800, (max tau/M)^2 = 1.000e-02 max_P |E_b - (11.12)| = 2.053e-06 = 0.021 * (tau/M)^4; max_P |E_b' - d(11.12)/dP| = 2.354e-07 E_b''(0): numeric +3.90559613e-03 closed +3.90434405e-03 rel.dev 3.207e-04 = 0.032 * (tau/M)^2 s3_1 numeric +9.9945154586e-01 closed +9.9945217202e-01 diff -6.26e-07 s3_2 numeric +9.9945154586e-01 closed +9.9945217202e-01 diff -6.26e-07 A1 numeric +1.0948993397e-03 closed +1.0956559528e-03 diff -7.57e-07 A2 numeric +1.0948993397e-03 closed +1.0956559528e-03 diff -7.57e-07 U numeric -2.4940068754e-03 closed -2.4987801902e-03 diff +4.77e-06 Ebind numeric -1.4011164735e-03 closed -1.4031242374e-03 diff +2.01e-06 Epp0 numeric +3.9055961250e-03 closed +3.9043440472e-03 diff +1.25e-06 m_b/(m_1+m_2): numeric +1.280214 (11.16) +1.280625; m_b-(m_1+m_2): numeric +5.604286e+01 -4 m_1 m_2 E_bind +5.604466e+01 === set B_unequal_R: {'M1': 1.0, 'M2': 1.7, 'tau1': 0.2, 'tau2': 0.12, 'w': [-0.01, 0.005, -0.004, 0.008]} max tau_i/M_i = 0.200 P=-0.30 E_b=2.696652071452 isolation=3.657e-03 in a band: False weight(r=0,11)=0.6441 Gamma=+1.000000 E_b'=-1.546134e-03 =-1.546134e-03 =-1.546134e-03 E_b''=+4.673251e-03 (11.9)=+4.673251e-03 P=-0.15 E_b=2.696475088077 isolation=3.603e-03 in a band: False weight(r=0,11)=0.6368 Gamma=+1.000000 E_b'=-8.001290e-04 =-8.001290e-04 =-8.001290e-04 E_b''=+5.213785e-03 (11.9)=+5.213785e-03 P=+0.00 E_b=2.696414739677 isolation=3.585e-03 in a band: False weight(r=0,11)=0.6344 Gamma=+1.000000 E_b'=+1.110223e-14 =-1.752348e-19 =+3.687192e-20 E_b''=+5.394417e-03 (11.9)=+5.394417e-03 P=+0.15 E_b=2.696475088077 isolation=3.603e-03 in a band: False weight(r=0,11)=0.6368 Gamma=+1.000000 E_b'=+8.001290e-04 =+8.001290e-04 =+8.001290e-04 E_b''=+5.213785e-03 (11.9)=+5.213785e-03 P=+0.30 E_b=2.696652071452 isolation=3.657e-03 in a band: False weight(r=0,11)=0.6441 Gamma=+1.000000 E_b'=+1.546134e-03 =+1.546134e-03 =+1.546134e-03 E_b''=+4.673251e-03 (11.9)=+4.673251e-03 max |deviation| over the P values: 2(a): dE/dM_i - 1.49e-12; tau_i dE/dtau_i - 1.56e-12; dE/dtheta - 6.98e-13; Euler residual 4.44e-15 2(b): E_b' - 8.50e-13; E_b' - 8.50e-13; - 7.35e-17 2(c): E_b'' (finite difference) - formula (11.9) 1.83e-11 3(a): coefficients of (11.10) against direct inversion (relative) 4.28e-14; root of (11.10) - E_b 4.88e-15 numerics: |E_b(L) - E_b(2L)| 3.11e-15; weight at |r|>L/2 9.86e-21; dependence on eta 6.22e-15; dense - banded 3.55e-15 closed forms (items 3, 4), regime R: |w11|/a = 0.825, (max tau/M)^2 = 4.000e-02 max_P |E_b - (11.12)| = 8.267e-06 = 0.005 * (tau/M)^4; max_P |E_b' - d(11.12)/dP| = 6.697e-07 E_b''(0): numeric +5.39441742e-03 closed +5.39148145e-03 rel.dev 5.446e-04 = 0.014 * (tau/M)^2 s3_1 numeric +9.9771238948e-01 closed +9.9771281374e-01 diff -4.24e-07 s3_2 numeric +9.9971332360e-01 closed +9.9971509098e-01 diff -1.77e-06 A1 numeric +4.5625732894e-03 closed +4.5743725191e-03 diff -1.18e-05 A2 numeric +9.7104322032e-04 closed +9.6869065111e-04 diff +2.35e-06 U numeric -6.3439164227e-03 closed -6.3649433780e-03 diff +2.10e-05 Ebind numeric -3.5852603230e-03 closed -3.5934117929e-03 diff +8.15e-06 Epp0 numeric +5.3944174247e-03 closed +5.3914814496e-03 diff +2.94e-06 m_b/(m_1+m_2): numeric +1.295838 (11.16) +1.296544; m_b-(m_1+m_2): numeric +4.232127e+01 -4 m_1 m_2 E_bind +4.232599e+01 === set C_above_band_R: {'M1': 1.0, 'M2': 0.8, 'tau1': 0.1, 'tau2': 0.15, 'w': [0.02, -0.005, 0.003, -0.01]} max tau_i/M_i = 0.187 P=-0.30 E_b=1.831346531561 isolation=1.274e-02 in a band: False weight(r=0,11)=0.9037 Gamma=+1.000000 E_b'=+8.867732e-04 =+8.867732e-04 =+8.867732e-04 E_b''=-2.629234e-03 (11.9)=-2.629234e-03 P=-0.15 E_b=1.831448371991 isolation=1.260e-02 in a band: False weight(r=0,11)=0.8994 Gamma=+1.000000 E_b'=+4.619112e-04 =+4.619112e-04 =+4.619112e-04 E_b''=-2.996418e-03 (11.9)=-2.996418e-03 P=+0.00 E_b=1.831483249143 isolation=1.255e-02 in a band: False weight(r=0,11)=0.8980 Gamma=+1.000000 E_b'=+1.850372e-15 =-1.604183e-19 =+5.735736e-19 E_b''=-3.121011e-03 (11.9)=-3.121011e-03 P=+0.15 E_b=1.831448371991 isolation=1.260e-02 in a band: False weight(r=0,11)=0.8994 Gamma=+1.000000 E_b'=-4.619112e-04 =-4.619112e-04 =-4.619112e-04 E_b''=-2.996418e-03 (11.9)=-2.996418e-03 P=+0.30 E_b=1.831346531561 isolation=1.274e-02 in a band: False weight(r=0,11)=0.9037 Gamma=+1.000000 E_b'=-8.867732e-04 =-8.867732e-04 =-8.867732e-04 E_b''=-2.629234e-03 (11.9)=-2.629234e-03 max |deviation| over the P values: 2(a): dE/dM_i - 1.04e-12; tau_i dE/dtau_i - 1.30e-12; dE/dtheta - 1.11e-12; Euler residual 1.78e-15 2(b): E_b' - 2.74e-12; E_b' - 2.74e-12; - 8.55e-17 2(c): E_b'' (finite difference) - formula (11.9) 1.26e-11 3(a): coefficients of (11.10) against direct inversion (relative) 4.97e-15; root of (11.10) - E_b 8.88e-16 numerics: |E_b(L) - E_b(2L)| 2.22e-16; weight at |r|>L/2 5.56e-28; dependence on eta 1.33e-15; dense - banded 4.44e-16 closed forms (items 3, 4), regime R: |w11|/a = 2.098, (max tau/M)^2 = 3.516e-02 max_P |E_b - (11.12)| = 2.030e-04 = 0.164 * (tau/M)^4; max_P |E_b' - d(11.12)/dP| = 1.488e-05 E_b''(0): numeric -3.12101105e-03 closed -3.17366056e-03 rel.dev 1.659e-02 = 0.472 * (tau/M)^2 s3_1 numeric +9.9650760383e-01 closed +9.9642448170e-01 diff +8.31e-05 s3_2 numeric +9.8795706438e-01 closed +9.8742981847e-01 diff +5.27e-04 A1 numeric +7.0534612009e-03 closed +7.1510366031e-03 diff -9.76e-05 A2 numeric +1.9596946814e-02 closed +2.0112290446e-02 diff -5.15e-04 U numeric +1.7959585791e-02 closed +1.8054602308e-02 diff -9.50e-05 Ebind numeric +3.1483249143e-02 closed +3.1686265833e-02 diff -2.03e-04 Epp0 numeric -3.1210110472e-03 closed -3.1736605620e-03 diff +5.26e-05 m_b/(m_1+m_2): numeric -2.363673 (11.16) -2.324461; m_b-(m_1+m_2): numeric -4.559645e+02 -4 m_1 m_2 E_bind -4.477618e+02 === set D_wide_R: {'M1': 1.3, 'M2': 1.0, 'tau1': 0.15, 'tau2': 0.2, 'w': [-0.004, 0.002, 0.001, 0.003]} max tau_i/M_i = 0.200 P=-0.30 E_b=2.299969840671 isolation=5.677e-04 in a band: False weight(r=0,11)=0.2780 Gamma=+1.000000 E_b'=-3.402251e-03 =-3.402251e-03 =-3.402251e-03 E_b''=+1.074662e-02 (11.9)=+1.074662e-02 P=-0.15 E_b=2.299583345287 isolation=5.522e-04 in a band: False weight(r=0,11)=0.2707 Gamma=+1.000000 E_b'=-1.734310e-03 =-1.734310e-03 =-1.734310e-03 E_b''=+1.141561e-02 (11.9)=+1.141561e-02 P=+0.00 E_b=2.299452860723 isolation=5.471e-04 in a band: False weight(r=0,11)=0.2684 Gamma=+1.000000 E_b'=+1.110223e-14 =-2.235218e-19 =+5.295572e-19 E_b''=+1.163513e-02 (11.9)=+1.163513e-02 P=+0.15 E_b=2.299583345287 isolation=5.522e-04 in a band: False weight(r=0,11)=0.2707 Gamma=+1.000000 E_b'=+1.734310e-03 =+1.734310e-03 =+1.734310e-03 E_b''=+1.141561e-02 (11.9)=+1.141561e-02 P=+0.30 E_b=2.299969840671 isolation=5.677e-04 in a band: False weight(r=0,11)=0.2780 Gamma=+1.000000 E_b'=+3.402251e-03 =+3.402251e-03 =+3.402251e-03 E_b''=+1.074662e-02 (11.9)=+1.074662e-02 max |deviation| over the P values: 2(a): dE/dM_i - 2.05e-12; tau_i dE/dtau_i - 1.96e-12; dE/dtheta - 2.07e-12; Euler residual 5.33e-15 2(b): E_b' - 5.04e-12; E_b' - 5.04e-12; - 8.46e-17 2(c): E_b'' (finite difference) - formula (11.9) 9.92e-11 3(a): coefficients of (11.10) against direct inversion (relative) 2.34e-13; root of (11.10) - E_b 1.78e-15 numerics: |E_b(L) - E_b(2L)| 4.44e-15; weight at |r|>L/2 4.85e-08; dependence on eta 4.44e-15; dense - banded 2.22e-15 closed forms (items 3, 4), regime R: |w11|/a = 0.279, (max tau/M)^2 = 4.000e-02 max_P |E_b - (11.12)| = 7.729e-07 = 0.000 * (tau/M)^4; max_P |E_b' - d(11.12)/dP| = 1.014e-06 E_b''(0): numeric +1.16351317e-02 closed +1.16355522e-02 rel.dev 3.615e-05 = 0.001 * (tau/M)^2 s3_1 numeric +9.9987712967e-01 closed +9.9987739882e-01 diff -2.69e-07 s3_2 numeric +9.9963191441e-01 closed +9.9963165158e-01 diff +2.63e-07 A1 numeric +3.1883906936e-04 closed +3.1876305792e-04 diff +7.60e-08 A2 numeric +7.3524924751e-04 closed +7.3669684496e-04 diff -1.45e-06 U numeric -1.0734105787e-03 closed -1.0756421628e-03 diff +2.23e-06 Ebind numeric -5.4713927724e-04 closed -5.4791221131e-04 diff +7.73e-07 Epp0 numeric +1.1635131656e-02 closed +1.1635552241e-02 diff -4.21e-07 m_b/(m_1+m_2): numeric +1.038281 (11.16) +1.038244; m_b-(m_1+m_2): numeric +3.168822e+00 -4 m_1 m_2 E_bind +3.161249e+00 === set E_order_one: {'M1': 1.0, 'M2': 1.3, 'tau1': 1.0, 'tau2': 0.8, 'w': [-0.6, 0.2, -0.15, 0.3]} max tau_i/M_i = 1.000 P=-0.30 E_b=2.020243065365 isolation=2.944e-01 in a band: False weight(r=0,11)=0.7017 Gamma=+1.000000 E_b'=-5.330716e-02 =-5.330716e-02 =-5.330716e-02 E_b''=+1.580211e-01 (11.9)=+1.580211e-01 P=-0.15 E_b=2.014120790386 isolation=2.896e-01 in a band: False weight(r=0,11)=0.6956 Gamma=+1.000000 E_b'=-2.776997e-02 =-2.776997e-02 =-2.776997e-02 E_b''=+1.801270e-01 (11.9)=+1.801270e-01 P=+0.00 E_b=2.012023935088 isolation=2.880e-01 in a band: False weight(r=0,11)=0.6936 Gamma=+1.000000 E_b'=-7.401487e-15 =-3.140157e-17 =+1.914190e-17 E_b''=+1.876436e-01 (11.9)=+1.876436e-01 P=+0.15 E_b=2.014120790386 isolation=2.896e-01 in a band: False weight(r=0,11)=0.6956 Gamma=+1.000000 E_b'=+2.776997e-02 =+2.776997e-02 =+2.776997e-02 E_b''=+1.801270e-01 (11.9)=+1.801270e-01 P=+0.30 E_b=2.020243065365 isolation=2.944e-01 in a band: False weight(r=0,11)=0.7017 Gamma=+1.000000 E_b'=+5.330716e-02 =+5.330716e-02 =+5.330716e-02 E_b''=+1.580211e-01 (11.9)=+1.580211e-01 max |deviation| over the P values: 2(a): dE/dM_i - 7.62e-13; tau_i dE/dtau_i - 1.52e-12; dE/dtheta - 1.15e-12; Euler residual 1.33e-15 2(b): E_b' - 1.74e-10; E_b' - 1.74e-10; - 3.54e-16 2(c): E_b'' (finite difference) - formula (11.9) 2.40e-10 3(a): coefficients of (11.10) against direct inversion (relative) 3.59e-15; root of (11.10) - E_b 1.33e-15 numerics: |E_b(L) - E_b(2L)| 8.88e-16; weight at |r|>L/2 2.26e-30; dependence on eta 8.88e-16; dense - banded 2.66e-15 closed forms (items 3, 4), control, outside R: |w11|/a = 1.608, (max tau/M)^2 = 1.000e+00 max_P |E_b - (11.12)| = 4.605e-02 = 0.046 * (tau/M)^4; max_P |E_b' - d(11.12)/dP| = 3.589e-03 E_b''(0): numeric +1.87643618e-01 closed +1.74198841e-01 rel.dev 7.718e-02 = 0.077 * (tau/M)^2 s3_1 numeric +8.5646967127e-01 closed +8.8201005947e-01 diff -2.55e-02 s3_2 numeric +9.3859089119e-01 closed +9.5531741897e-01 diff -1.67e-02 A1 numeric +2.2543811170e-01 closed +2.3597988106e-01 diff -1.05e-02 A2 numeric +1.2495039252e-01 closed +1.1617471068e-01 diff +8.78e-03 U numeric -4.1500239894e-01 closed -5.0953161098e-01 diff +9.45e-02 Ebind numeric -2.8797606491e-01 closed -3.3345431511e-01 diff +4.55e-02 Epp0 numeric +1.8764361764e-01 closed +1.7419884136e-01 diff +1.34e-02 m_b/(m_1+m_2): numeric +1.758104 (11.16) +1.893795; m_b-(m_1+m_2): numeric +2.298001e+00 -4 m_1 m_2 E_bind +2.339806e+00 === epsilon scan on set B_unequal_R (tau -> eps*tau, w -> eps^2*w) eps= 1.000 E_b-(11.12): P=0 +8.1515e-06, P=0.3 +8.2675e-06; rel.dev E_b''(0) +5.4456e-04; -(11.14): -4.2426e-07, -1.7674e-06 eps= 0.500 E_b-(11.12): P=0 +5.0943e-07, P=0.3 +5.1674e-07; rel.dev E_b''(0) +1.3708e-04; -(11.14): -2.7203e-08, -1.1020e-07 ratios to the previous eps: dE_P0 16.00, dE_P03 16.00, rel_dEpp0 3.97, d_s3_1 15.60, d_s3_2 16.04 eps= 0.250 E_b-(11.12): P=0 +3.1839e-08, P=0.3 +3.2297e-08; rel.dev E_b''(0) +3.4328e-05; -(11.14): -1.7109e-09, -6.8832e-09 ratios to the previous eps: dE_P0 16.00, dE_P03 16.00, rel_dEpp0 3.99, d_s3_1 15.90, d_s3_2 16.01 eps= 0.125 E_b-(11.12): P=0 +1.9899e-09, P=0.3 +2.0186e-09; rel.dev E_b''(0) +8.5858e-06; -(11.14): -1.0710e-10, -4.3014e-10 ratios to the previous eps: dE_P0 16.00, dE_P03 16.00, rel_dEpp0 4.00, d_s3_1 15.97, d_s3_2 16.00 expected for halving eps: 16 for E_b and (next order eps^4), 4 for the relative deviation of E_b''(0) === new in v2, I3: reading statistics of the clock of a part, (11.19), (11.25), (11.26) start: bound eigenvector of the level 0 on every level, clock label in |0>, total label P=0 sharp, no records; readings at lambda = t*lambda_0, t = 0.25, 0.5, ..., 2N; deviation = max over a and t of |q - P_N(a; Gamma*t)| set E_order_one, clock of part 1, N=4, spread D=0.0600 (D/g0=0.208, D/m=0.0600, lambda_0=78.54) Gamma: level 0 (v1) 0.8564696713; mean level (11.19) 0.8621903823; first order (11.25) 0.8623337320; least-squares fit 0.8621456724 d2E_b/dM^2 at m: spectral sum +1.95468691e-01; closed form of (11.26) +1.72184621e-01; closed (11.20) at the mean level 0.8870173806; closed (11.26) 0.8871755981 deviation of the statistics: Gamma at the mean level 9.468e-04 (t<=N: 3.598e-04); first order 2.645e-03; Gamma of the level 0 7.098e-02 (t<=N: 3.790e-02) set E_order_one, clock of part 1, N=4, spread D=0.0300 (D/g0=0.104, D/m=0.0300, lambda_0=157.08) Gamma: level 0 (v1) 0.8564696713; mean level (11.19) 0.8593655981; first order (11.25) 0.8594017016; least-squares fit 0.8593541025 d2E_b/dM^2 at m: spectral sum +1.95468691e-01; closed form of (11.26) +1.72184621e-01; closed (11.20) at the mean level 0.8845527554; closed (11.26) 0.8845928288 deviation of the statistics: Gamma at the mean level 2.476e-04 (t<=N: 9.235e-05); first order 6.867e-04; Gamma of the level 0 3.606e-02 (t<=N: 1.923e-02) ratios to the previous spread: dev_mean_level 3.82, dev_level0 1.97, dev_first_order 3.85 set E_order_one, clock of part 1, N=4, spread D=0.0150 (D/g0=0.052, D/m=0.0150, lambda_0=314.16) Gamma: level 0 (v1) 0.8564696713; mean level (11.19) 0.8579266274; first order (11.25) 0.8579356864; least-squares fit 0.8579237667 d2E_b/dM^2 at m: spectral sum +1.95468691e-01; closed form of (11.26) +1.72184621e-01; closed (11.20) at the mean level 0.8832913598; closed (11.26) 0.8833014441 deviation of the statistics: Gamma at the mean level 6.077e-05 (t<=N: 2.322e-05); first order 1.707e-04; Gamma of the level 0 1.816e-02 (t<=N: 9.686e-03) ratios to the previous spread: dev_mean_level 4.07, dev_level0 1.99, dev_first_order 4.02 set B_unequal_R, clock of part 1, N=4, spread D=0.1000 (D/g0=27.860, D/m=0.1000, lambda_0=47.12) Gamma: level 0 (v1) 0.9977123895; mean level (11.19) 0.9978093968; first order (11.25) 0.9978119033; least-squares fit 0.9978328312 d2E_b/dM^2 at m: spectral sum +1.99027585e-03; closed form of (11.26) +1.99577457e-03; closed (11.20) at the mean level 0.9978100720; closed (11.26) 0.9978126025 deviation of the statistics: Gamma at the mean level 4.686e-04 (t<=N: 4.128e-04); first order 4.707e-04; Gamma of the level 0 1.426e-03 (t<=N: 9.115e-04) set B_unequal_R, clock of part 1, N=4, spread D=0.0500 (D/g0=13.930, D/m=0.0500, lambda_0=94.25) Gamma: level 0 (v1) 0.9977123895; mean level (11.19) 0.9977615160; first order (11.25) 0.9977621464; least-squares fit 0.9977620037 d2E_b/dM^2 at m: spectral sum +1.99027585e-03; closed form of (11.26) +1.99577457e-03; closed (11.20) at the mean level 0.9977620716; closed (11.26) 0.9977627081 deviation of the statistics: Gamma at the mean level 1.307e-04 (t<=N: 1.251e-04); first order 1.309e-04; Gamma of the level 0 5.874e-04 (t<=N: 3.517e-04) ratios to the previous spread: dev_mean_level 3.59, dev_level0 2.43, dev_first_order 3.60 set B_unequal_R, clock of part 1, N=4, spread D=0.0250 (D/g0=6.965, D/m=0.0250, lambda_0=188.50) Gamma: level 0 (v1) 0.9977123895; mean level (11.19) 0.9977371099; first order (11.25) 0.9977372679; least-squares fit 0.9977368502 d2E_b/dM^2 at m: spectral sum +1.99027585e-03; closed form of (11.26) +1.99577457e-03; closed (11.20) at the mean level 0.9977376013; closed (11.26) 0.9977377609 deviation of the statistics: Gamma at the mean level 3.259e-05 (t<=N: 3.259e-05); first order 3.261e-05; Gamma of the level 0 3.021e-04 (t<=N: 1.408e-04) ratios to the previous spread: dev_mean_level 4.01, dev_level0 1.94, dev_first_order 4.01 set B_unequal_R, clock of part 2, N=5, spread D=0.1000 (D/g0=27.860, D/m=0.0588, lambda_0=50.27) Gamma: level 0 (v1) 0.9997133236; mean level (11.19) 0.9997276451; first order (11.25) 0.9997282035; least-squares fit 0.9997277090 d2E_b/dM^2 at m: spectral sum +2.97598087e-04; closed form of (11.26) +2.95174829e-04; closed (11.20) at the mean level 0.9997292967; closed (11.26) 0.9997298497 deviation of the statistics: Gamma at the mean level 7.255e-06 (t<=N: 5.690e-06); first order 1.305e-05; Gamma of the level 0 2.199e-04 (t<=N: 1.083e-04) set B_unequal_R, clock of part 2, N=5, spread D=0.0500 (D/g0=13.930, D/m=0.0294, lambda_0=100.53) Gamma: level 0 (v1) 0.9997133236; mean level (11.19) 0.9997206216; first order (11.25) 0.9997207635; least-squares fit 0.9997205698 d2E_b/dM^2 at m: spectral sum +2.97598087e-04; closed form of (11.26) +2.95174829e-04; closed (11.20) at the mean level 0.9997223298; closed (11.26) 0.9997224704 deviation of the statistics: Gamma at the mean level 1.889e-06 (t<=N: 1.889e-06); first order 3.433e-06; Gamma of the level 0 1.114e-04 (t<=N: 5.357e-05) ratios to the previous spread: dev_mean_level 3.84, dev_level0 1.97, dev_first_order 3.80 set B_unequal_R, clock of part 2, N=5, spread D=0.0250 (D/g0=6.965, D/m=0.0147, lambda_0=201.06) Gamma: level 0 (v1) 0.9997133236; mean level (11.19) 0.9997170078; first order (11.25) 0.9997170436; least-squares fit 0.9997169943 d2E_b/dM^2 at m: spectral sum +2.97598087e-04; closed form of (11.26) +2.95174829e-04; closed (11.20) at the mean level 0.9997187452; closed (11.26) 0.9997187807 deviation of the statistics: Gamma at the mean level 4.854e-07 (t<=N: 4.854e-07); first order 8.020e-07; Gamma of the level 0 5.664e-05 (t<=N: 2.673e-05) ratios to the previous spread: dev_mean_level 3.89, dev_level0 1.97, dev_first_order 4.28 expected for halving the spread at fixed lambda/lambda_0: 4 with Gamma at the mean level, 2 with Gamma of the level 0 === new in v2, I4: order of the contribution of w22 (set B, tau -> eps*tau, w -> eps^2*w) eps= 2.000 E_b(w22) - E_b(w22=0) = +2.867296e-07; w22*|<22|psi(0)>|^2 = +2.867347e-07; |<22|psi(0)>|^2 = 8.960461e-06 eps= 1.000 E_b(w22) - E_b(w22=0) = +4.444423e-09; w22*|<22|psi(0)>|^2 = +4.444423e-09; |<22|psi(0)>|^2 = 5.555528e-07; ratios to the previous eps: dE 64.51, weight 16.13 eps= 0.500 E_b(w22) - E_b(w22=0) = +6.930323e-11; w22*|<22|psi(0)>|^2 = +6.930409e-11; |<22|psi(0)>|^2 = 3.465205e-08; ratios to the previous eps: dE 64.13, weight 16.03 eps= 0.250 E_b(w22) - E_b(w22=0) = +1.080025e-12; w22*|<22|psi(0)>|^2 = +1.082329e-12; |<22|psi(0)>|^2 = 2.164659e-09; ratios to the previous eps: dE 64.17, weight 16.01 expected for halving eps: 64 for the contribution of w22 to E_b (order eps^6), 16 for the weight (order eps^4) === new in v2, I5: equal data (a_1 = a_2) at P = pi/2, where j(P) = 0 (set A, tau -> eps*tau, w -> eps^2*w) eps= 1.000 E_b = 2.001000623470 M_1+M_2+a+w11 = 2.001000000000 difference +6.235e-07; 1 - weight(r=0,11) = 2.495e-03; weight of 11 at r != 0: 3.044e-07 eps= 0.500 E_b = 2.000250039039 M_1+M_2+a+w11 = 2.000250000000 difference +3.904e-08; 1 - weight(r=0,11) = 6.247e-04; weight of 11 at r != 0: 1.906e-08; ratios: dE 15.97, 1-weight 3.99 eps= 0.250 E_b = 2.000062502441 M_1+M_2+a+w11 = 2.000062500000 difference +2.441e-09; 1 - weight(r=0,11) = 1.562e-04; weight of 11 at r != 0: 1.192e-09; ratios: dE 15.99, 1-weight 4.00 expected: psi_P = delta_{r,0}|11> up to O(eps) (1 - weight of order eps^2), E_b = M_1+M_2+a+w11 up to O(eps^4) === new in v2, I2: twofold bound levels (on-site binding W0 = -G(E0,P0)^(-1) on span(|11>,|22>)) set X_P0_0.3: M = 1.0, 0.7; tau = 0.5, 0.35; P0 = 0.3; W0 on (11,22) = [[-0.870998, 0.169999], [0.169999, 2.488876]]; bands {'+1+1': (1.7046, 1.8965), '+1-1': (0.2832, 0.3515), '-1+1': (-0.3515, -0.2832), '-1-1': (-1.8965, -1.7046)} level 1.000000000000, splitting of the two eigenvalues 2.11e-15, distance to the next eigenvalue 0.5368, weight at |r|>L/2 1.3e-30 M1 : eigenvalues of Pi dH Pi -0.628449130, +0.893076371; slopes of the branches (finite differences) -0.628449130, +0.893076371; max dev 1.8e-12; overlap with the branch vectors >= 0.999997 M2 : eigenvalues of Pi dH Pi -0.919383668, +0.837955656; slopes of the branches (finite differences) -0.919383668, +0.837955656; max dev 8.3e-13; overlap with the branch vectors >= 0.999999 tau1 : eigenvalues of Pi dH Pi +0.137013648, +0.601738268; slopes of the branches (finite differences) +0.137013648, +0.601738268; max dev 8.2e-12; overlap with the branch vectors >= 0.999999 tau2 : eigenvalues of Pi dH Pi -0.156239994, +0.533638710; slopes of the branches (finite differences) -0.156239994, +0.533638710; max dev 1.1e-12; overlap with the branch vectors >= 1.000000 theta: eigenvalues of Pi dH Pi -0.726064561, +2.016971420; slopes of the branches (finite differences) -0.726064561, +2.016971420; max dev 4.6e-12; overlap with the branch vectors >= 0.999996 P : eigenvalues of Pi dH Pi -0.031846375, +0.056264981; slopes of the branches (finite differences) -0.031846375, +0.056264981; max dev 2.9e-10; overlap with the branch vectors >= 0.999955 E'' of the branches: (11.24) -0.10636246, +0.17322408; finite differences -0.10636246, +0.17322408; max dev 1.3e-10 (11.9) with Q_P = 1 - |psi>|^2 = 6.549e-04 over E' - E = -2.109e-15 every eigenvector (5 random vectors of the level): residual of (11.6) 1.6e-15; - 1.1e-16; ||Pi (V1 - V2) Pi|| 1.6e-16; - weighted mean of the branch slopes (M1, theta) 4.1e-12 ||^2: M2 0.9912, tau1 0.2769, tau2 0.7272, theta 0.0013, P 0.5106 in the two branch vectors of M1: +2.013542, -0.722635 against the slopes of the branches of theta: -0.726065, +2.016971 set X_P0_0: M = 1.0, 0.7; tau = 0.5, 0.35; P0 = 0.0; W0 on (11,22) = [[-0.869086, 0.177794], [0.177794, 2.488618]]; bands {'+1+1': (1.7, 1.9007), '+1-1': (0.3, 0.3354), '-1+1': (-0.3354, -0.3), '-1-1': (-1.9007, -1.7)} level 1.000000000000, splitting of the two eigenvalues 3.55e-15, distance to the next eigenvalue 0.5397, weight at |r|>L/2 3.2e-30 M1 : eigenvalues of Pi dH Pi -0.631196110, +0.895473254; slopes of the branches (finite differences) -0.631196110, +0.895473254; max dev 1.8e-12; overlap with the branch vectors >= 0.999997 M2 : eigenvalues of Pi dH Pi -0.918901736, +0.845071613; slopes of the branches (finite differences) -0.918901736, +0.845071613; max dev 1.2e-12; overlap with the branch vectors >= 0.999999 tau1 : eigenvalues of Pi dH Pi +0.120869467, +0.605304949; slopes of the branches (finite differences) +0.120869467, +0.605304949; max dev 2.1e-12; overlap with the branch vectors >= 0.999999 tau2 : eigenvalues of Pi dH Pi -0.179004420, +0.538130054; slopes of the branches (finite differences) -0.179004420, +0.538130054; max dev 7.5e-13; overlap with the branch vectors >= 1.000000 theta: eigenvalues of Pi dH Pi -0.725682673, +2.024305435; slopes of the branches (finite differences) -0.725682673, +2.024305435; max dev 3.7e-12; overlap with the branch vectors >= 0.999996 P : eigenvalues of Pi dH Pi -0.000000000, +0.000000000; slopes of the branches (finite differences) +0.000000000, -0.000000000; max dev 1.8e-14; overlap with the branch vectors >= 0.999952; branch vectors from the second order E'' of the branches: (11.24) -0.10630610, +0.19294341; finite differences -0.10630610, +0.19294341; max dev 4.1e-10 (11.9) with Q_P = 1 - |psi>|^2 = 9.382e-35 over E' - E = -3.553e-15 every eigenvector (5 random vectors of the level): residual of (11.6) 1.6e-15; - 3.5e-17; ||Pi (V1 - V2) Pi|| 6.4e-17; - weighted mean of the branch slopes (M1, theta) 2.9e-12 ||^2: M2 0.9904, tau1 0.2843, tau2 0.7171, theta 0.0014, P 0.5112 in the two branch vectors of M1: +2.020503, -0.721880 against the slopes of the branches of theta: -0.725683, +2.024305 === new in v2, I1: the operator C_bod = of (11.21), small system in the places (dimension 64 x 3) listing A: the terms of the question; listing B: h*sigma3 of b_1, a constant and a part of the motion term of b_2 listed in the record terms; = -0.6021 for the medium that is not centred; lambda = 3 kappa=0.040 centred : ||C_B - C_A|| 1.3e-15; ||C_bod(B) - C_bod(A)|| 5.2e-16; ||C_bod - C_pair(A)|| 1.339e-15 (minus the gradient: 1.3e-15); ||C_bod - C_pair(B)|| 2.589; view against evolution by C_bod 4.141e-02, by C_pair(A) 4.141e-02, by C_pair(B) 1.025 kappa=0.040 not_centred: ||C_B - C_A|| 1.5e-15; ||C_bod(B) - C_bod(A)|| 1.3e-15; ||C_bod - C_pair(A)|| 6.534e-01 (minus the gradient: 1.3e-15); ||C_bod - C_pair(B)|| 2.491; view against evolution by C_bod 4.147e-02, by C_pair(A) 1.526e-01, by C_pair(B) 1.045 kappa=0.020 centred : ||C_B - C_A|| 1.3e-15; ||C_bod(B) - C_bod(A)|| 1.0e-15; ||C_bod - C_pair(A)|| 1.298e-15 (minus the gradient: 1.3e-15); ||C_bod - C_pair(B)|| 2.589; view against evolution by C_bod 1.052e-02, by C_pair(A) 1.052e-02, by C_pair(B) 1.034 kappa=0.020 not_centred: ||C_B - C_A|| 1.4e-15; ||C_bod(B) - C_bod(A)|| 9.8e-16; ||C_bod - C_pair(A)|| 3.267e-01 (minus the gradient: 1.5e-15); ||C_bod - C_pair(B)|| 2.520; view against evolution by C_bod 1.053e-02, by C_pair(A) 7.442e-02, by C_pair(B) 1.044 kappa=0.010 centred : ||C_B - C_A|| 1.4e-15; ||C_bod(B) - C_bod(A)|| 8.3e-16; ||C_bod - C_pair(A)|| 1.295e-15 (minus the gradient: 1.3e-15); ||C_bod - C_pair(B)|| 2.589; view against evolution by C_bod 2.642e-03, by C_pair(A) 2.642e-03, by C_pair(B) 1.037 kappa=0.010 not_centred: ||C_B - C_A|| 1.3e-15; ||C_bod(B) - C_bod(A)|| 1.2e-15; ||C_bod - C_pair(A)|| 1.633e-01 (minus the gradient: 1.8e-15); ||C_bod - C_pair(B)|| 2.549; view against evolution by C_bod 2.643e-03, by C_pair(A) 3.695e-02, by C_pair(B) 1.041 expected for halving kappa: the view against the evolution by C_bod falls by 4 (second order); by the listed C_pair(A) with a medium that is not centred it falls by 2 === new in v3, I3: wide pair at fixed epsilon (data of set B_unequal_R, w11 = -u*a or +u*a, the other three w fixed) level on the infinite chain: root of F_+ of (11.10), mean over 32768 values of q; x = E_b - (M_1+M_2); E_b''(0) from 5-point differences of the roots, step 0.01 cross-check of the root against the dense diagonalization (L = 120), eps' = 1: sign -1, u = 1.0: root - dense = -3.00e-15 sign -1, u = 0.5: root - dense = +9.69e-16 sign +1, u = 1.0: root - dense = -8.28e-15 sign +1, u = 0.5: root - dense = -2.88e-15 eps' = 1.0 (max tau_i/M_i = 0.20), w11 < 0, level below the band: u= 2.000000 eps^2/u^2=1.000e-02 P=0: isolation 1.4920e-02, remainder x-(11.12) +5.823e-05 (over isolation 3.90e-03), rel.dev E_bind -3.89e-03, wide form -6.24e-01, E_b''(0) +4.56e-03; m_b-(m_1+m_2) = 1.753737e+02 against -4 m_1 m_2 E_bind (rel.dev -4.36e-03) and the leading function (rel.dev -8.22e-03); P=pi/2: isolation 1.7532e-02, remainder +6.498e-05 (over isolation 3.71e-03), (11.12) minus the exact edge -1.760e-02 u= 1.000000 eps^2/u^2=4.000e-02 P=0: isolation 5.0061e-03, remainder x-(11.12) +1.319e-05 (over isolation 2.64e-03), rel.dev E_bind -2.63e-03, wide form -2.10e-01, E_b''(0) +9.48e-04; m_b-(m_1+m_2) = 5.906384e+01 against -4 m_1 m_2 E_bind (rel.dev -6.08e-04) and the leading function (rel.dev -3.24e-03); P=pi/2: isolation 6.5542e-03, remainder +1.397e-05 (over isolation 2.13e-03), (11.12) minus the exact edge -6.568e-03 u= 0.500000 eps^2/u^2=1.600e-01 P=0: isolation 1.4282e-03, remainder x-(11.12) +2.076e-06 (over isolation 1.45e-03), rel.dev E_bind -1.45e-03, wide form -6.06e-02, E_b''(0) +9.59e-05; m_b-(m_1+m_2) = 1.687008e+01 against -4 m_1 m_2 E_bind (rel.dev +5.44e-04) and the leading function (rel.dev -9.08e-04); P=pi/2: isolation 2.0551e-03, remainder -8.013e-07 (over isolation 3.90e-04), (11.12) minus the exact edge -2.054e-03 u= 0.250000 eps^2/u^2=6.400e-01 P=0: isolation 3.7267e-04, remainder x-(11.12) +2.709e-07 (over isolation 7.27e-04), rel.dev E_bind -7.27e-04, wide form -1.61e-02, E_b''(0) +1.33e-06; m_b-(m_1+m_2) = 4.402540e+00 against -4 m_1 m_2 E_bind (rel.dev +6.82e-04) and the leading function (rel.dev -4.46e-05); P=pi/2: isolation 5.6088e-04, remainder -4.044e-06 (over isolation 7.21e-03), (11.12) minus the exact edge -5.568e-04 u= 0.125000 eps^2/u^2=2.560e+00 P=0: isolation 9.4268e-05, remainder x-(11.12) +3.369e-08 (over isolation 3.57e-04), rel.dev E_bind -3.57e-04, wide form -4.25e-03, E_b''(0) -1.72e-06; m_b-(m_1+m_2) = 1.113538e+00 against -4 m_1 m_2 E_bind (rel.dev +5.80e-04) and the leading function (rel.dev +2.23e-04); P=pi/2: isolation 1.4386e-04, remainder -4.902e-06 (over isolation 3.41e-02), (11.12) minus the exact edge -1.390e-04 u= 0.062500 eps^2/u^2=1.024e+01 P=0: isolation 2.3640e-05, remainder x-(11.12) +4.171e-09 (over isolation 1.76e-04), rel.dev E_bind -1.76e-04, wide form -1.15e-03, E_b''(0) -5.85e-07; m_b-(m_1+m_2) = 2.792169e-01 against -4 m_1 m_2 E_bind (rel.dev +4.77e-04) and the leading function (rel.dev +3.00e-04); P=pi/2: isolation 3.6200e-05, remainder -5.162e-06 (over isolation 1.43e-01), (11.12) minus the exact edge -3.104e-05 u= 0.031250 eps^2/u^2=4.096e+01 P=0: isolation 5.9149e-06, remainder x-(11.12) +5.181e-10 (over isolation 8.76e-05), rel.dev E_bind -8.76e-05, wide form -3.32e-04, E_b''(0) -1.57e-07; m_b-(m_1+m_2) = 6.985683e-02 against -4 m_1 m_2 E_bind (rel.dev +4.10e-04) and the leading function (rel.dev +3.23e-04); P=pi/2: isolation 9.0639e-06, remainder -5.236e-06 (over isolation 5.78e-01), (11.12) minus the exact edge -3.828e-06 u= 0.015625 eps^2/u^2=1.638e+02 P=0: isolation 1.4791e-06, remainder x-(11.12) +6.452e-11 (over isolation 4.36e-05), rel.dev E_bind -4.36e-05, wide form -1.05e-04, E_b''(0) -4.00e-08; m_b-(m_1+m_2) = 1.746749e-02 against -4 m_1 m_2 E_bind (rel.dev +3.71e-04) and the leading function (rel.dev +3.27e-04); P=pi/2: isolation 2.2667e-06, remainder -5.256e-06 (over isolation 2.32e+00), (11.12) minus the exact edge +2.989e-06 u= 0.007812 eps^2/u^2=6.554e+02 P=0: isolation 3.6979e-07, remainder x-(11.12) +8.050e-12 (over isolation 2.18e-05), rel.dev E_bind -2.18e-05, wide form -3.70e-05, E_b''(0) -9.77e-09; m_b-(m_1+m_2) = 4.367040e-03 against -4 m_1 m_2 E_bind (rel.dev +3.42e-04) and the leading function (rel.dev +3.20e-04); P=pi/2: isolation 5.6672e-07, remainder -5.261e-06 (over isolation 9.28e+00), (11.12) minus the exact edge +4.694e-06 eps' = 1.0, w11 > 0, level above the band, P = 0: u= 2.000000 eps^2/u^2=1.000e-02 isolation 1.4909e-02 (leading 1.4978e-02, rel.dev -4.62e-03), remainder x-(11.12) -2.706e-04 (over isolation 1.81e-02), E_bind-2a = +1.4708e-02 against w11^2/(2a) = 2.4235e-02 u= 1.000000 eps^2/u^2=4.000e-02 isolation 4.9971e-03 (leading 5.0193e-03, rel.dev -4.41e-03), remainder x-(11.12) -2.235e-04 (over isolation 4.47e-02), E_bind-2a = +4.7958e-03 against w11^2/(2a) = 6.0588e-03 u= 0.500000 eps^2/u^2=1.600e-01 isolation 1.4238e-03 (leading 1.4303e-03, rel.dev -4.52e-03), remainder x-(11.12) -2.078e-04 (over isolation 1.46e-01), E_bind-2a = +1.2225e-03 against w11^2/(2a) = 1.5147e-03 u= 0.250000 eps^2/u^2=6.400e-01 isolation 3.7115e-04 (leading 3.7294e-04, rel.dev -4.78e-03), remainder x-(11.12) -2.031e-04 (over isolation 5.47e-01), E_bind-2a = +1.6979e-04 against w11^2/(2a) = 3.7868e-04 u= 0.125000 eps^2/u^2=2.560e+00 isolation 9.3834e-05 (leading 9.4302e-05, rel.dev -4.96e-03), remainder x-(11.12) -2.018e-04 (over isolation 2.15e+00), E_bind-2a = -1.0753e-04 against w11^2/(2a) = 9.4669e-05 u= 0.062500 eps^2/u^2=1.024e+01 isolation 2.3526e-05 (leading 2.3644e-05, rel.dev -4.98e-03), remainder x-(11.12) -2.015e-04 (over isolation 8.56e+00), E_bind-2a = -1.7783e-04 against w11^2/(2a) = 2.3667e-05 u= 0.031250 eps^2/u^2=4.096e+01 isolation 5.8869e-06 (leading 5.9154e-06, rel.dev -4.81e-03), remainder x-(11.12) -2.014e-04 (over isolation 3.42e+01), E_bind-2a = -1.9547e-04 against w11^2/(2a) = 5.9168e-06 u= 0.015625 eps^2/u^2=1.638e+02 isolation 1.4727e-06 (leading 1.4791e-06, rel.dev -4.33e-03), remainder x-(11.12) -2.014e-04 (over isolation 1.37e+02), E_bind-2a = -1.9989e-04 against w11^2/(2a) = 1.4792e-06 u= 0.007812 eps^2/u^2=6.554e+02 isolation 3.6856e-07 (leading 3.6980e-07, rel.dev -3.33e-03), remainder x-(11.12) -2.014e-04 (over isolation 5.46e+02), E_bind-2a = -2.0099e-04 against w11^2/(2a) = 3.6980e-07 eps' = 0.5 (max tau_i/M_i = 0.10), w11 < 0, level below the band: u= 2.000000 eps^2/u^2=2.500e-03 P=0: isolation 3.7409e-03, remainder x-(11.12) +3.631e-06 (over isolation 9.71e-04), rel.dev E_bind -9.70e-04, wide form -6.20e-01, E_b''(0) +1.15e-03; m_b-(m_1+m_2) = 7.058394e+02 against -4 m_1 m_2 E_bind (rel.dev -1.11e-03) and the leading function (rel.dev -2.07e-03); P=pi/2: isolation 4.3963e-03, remainder +4.056e-06 (over isolation 9.23e-04), (11.12) minus the exact edge -4.400e-03 u= 1.000000 eps^2/u^2=1.000e-02 P=0: isolation 1.2540e-03, remainder x-(11.12) +8.245e-07 (over isolation 6.58e-04), rel.dev E_bind -6.57e-04, wide form -2.08e-01, E_b''(0) +2.39e-04; m_b-(m_1+m_2) = 2.368291e+02 against -4 m_1 m_2 E_bind (rel.dev -1.58e-04) and the leading function (rel.dev -8.15e-04); P=pi/2: isolation 1.6421e-03, remainder +8.743e-07 (over isolation 5.32e-04), (11.12) minus the exact edge -1.643e-03 u= 0.500000 eps^2/u^2=4.000e-02 P=0: isolation 3.5744e-04, remainder x-(11.12) +1.297e-07 (over isolation 3.63e-04), rel.dev E_bind -3.63e-04, wide form -5.94e-02, E_b''(0) +2.42e-05; m_b-(m_1+m_2) = 6.752618e+01 against -4 m_1 m_2 E_bind (rel.dev +1.33e-04) and the leading function (rel.dev -2.29e-04); P=pi/2: isolation 5.1460e-04, remainder -5.014e-08 (over isolation 9.74e-05), (11.12) minus the exact edge -5.146e-04 u= 0.250000 eps^2/u^2=1.600e-01 P=0: isolation 9.3217e-05, remainder x-(11.12) +1.692e-08 (over isolation 1.82e-04), rel.dev E_bind -1.82e-04, wide form -1.56e-02, E_b''(0) +3.82e-07; m_b-(m_1+m_2) = 1.761072e+01 against -4 m_1 m_2 E_bind (rel.dev +1.69e-04) and the leading function (rel.dev -1.28e-05); P=pi/2: isolation 1.4045e-04, remainder -2.532e-07 (over isolation 1.80e-03), (11.12) minus the exact edge -1.402e-04 u= 0.125000 eps^2/u^2=6.400e-01 P=0: isolation 2.3573e-05, remainder x-(11.12) +2.104e-09 (over isolation 8.92e-05), rel.dev E_bind -8.92e-05, wide form -3.98e-03, E_b''(0) -4.18e-07; m_b-(m_1+m_2) = 4.453399e+00 against -4 m_1 m_2 E_bind (rel.dev +1.43e-04) and the leading function (rel.dev +5.41e-05); P=pi/2: isolation 3.6032e-05, remainder -3.069e-07 (over isolation 8.52e-03), (11.12) minus the exact edge -3.572e-05 u= 0.062500 eps^2/u^2=2.560e+00 P=0: isolation 5.9108e-06, remainder x-(11.12) +2.604e-10 (over isolation 4.41e-05), rel.dev E_bind -4.41e-05, wide form -1.02e-03, E_b''(0) -1.43e-07; m_b-(m_1+m_2) = 1.116614e+00 against -4 m_1 m_2 E_bind (rel.dev +1.17e-04) and the leading function (rel.dev +7.34e-05); P=pi/2: isolation 9.0688e-06, remainder -3.232e-07 (over isolation 3.56e-02), (11.12) minus the exact edge -8.746e-06 u= 0.031250 eps^2/u^2=1.024e+01 P=0: isolation 1.4788e-06, remainder x-(11.12) +3.234e-11 (over isolation 2.19e-05), rel.dev E_bind -2.19e-05, wide form -2.66e-04, E_b''(0) -3.83e-08; m_b-(m_1+m_2) = 2.793591e-01 against -4 m_1 m_2 E_bind (rel.dev +1.00e-04) and the leading function (rel.dev +7.85e-05); P=pi/2: isolation 2.2710e-06, remainder -3.278e-07 (over isolation 1.44e-01), (11.12) minus the exact edge -1.943e-06 u= 0.015625 eps^2/u^2=4.096e+01 P=0: isolation 3.6977e-07, remainder x-(11.12) +4.027e-12 (over isolation 1.09e-05), rel.dev E_bind -1.09e-05, wide form -7.19e-05, E_b''(0) -9.50e-09; m_b-(m_1+m_2) = 6.985252e-02 against -4 m_1 m_2 E_bind (rel.dev +8.87e-05) and the leading function (rel.dev +7.78e-05); P=pi/2: isolation 5.6798e-07, remainder -3.291e-07 (over isolation 5.79e-01), (11.12) minus the exact edge -2.389e-07 u= 0.007812 eps^2/u^2=1.638e+02 P=0: isolation 9.2448e-08, remainder x-(11.12) +5.024e-13 (over isolation 5.43e-06), rel.dev E_bind -5.43e-06, wide form -2.07e-05, E_b''(0) -2.09e-09; m_b-(m_1+m_2) = 1.746377e-02 against -4 m_1 m_2 E_bind (rel.dev +7.40e-05) and the leading function (rel.dev +6.86e-05); P=pi/2: isolation 1.4201e-07, remainder -3.294e-07 (over isolation 2.32e+00), (11.12) minus the exact edge +1.874e-07 eps' = 0.5, w11 > 0, level above the band, P = 0: u= 2.000000 eps^2/u^2=2.500e-03 isolation 3.7401e-03 (leading 3.7446e-03, rel.dev -1.19e-03), remainder x-(11.12) -1.723e-05 (over isolation 4.61e-03), E_bind-2a = +3.7273e-03 against w11^2/(2a) = 6.0588e-03 u= 1.000000 eps^2/u^2=1.000e-02 isolation 1.2534e-03 (leading 1.2548e-03, rel.dev -1.14e-03), remainder x-(11.12) -1.419e-05 (over isolation 1.13e-02), E_bind-2a = +1.2406e-03 against w11^2/(2a) = 1.5147e-03 u= 0.500000 eps^2/u^2=4.000e-02 isolation 3.5716e-04 (leading 3.5757e-04, rel.dev -1.16e-03), remainder x-(11.12) -1.318e-05 (over isolation 3.69e-02), E_bind-2a = +3.4439e-04 against w11^2/(2a) = 3.7868e-04 u= 0.250000 eps^2/u^2=1.600e-01 isolation 9.3119e-05 (leading 9.3234e-05, rel.dev -1.24e-03), remainder x-(11.12) -1.288e-05 (over isolation 1.38e-01), E_bind-2a = +8.0352e-05 against w11^2/(2a) = 9.4669e-05 u= 0.125000 eps^2/u^2=6.400e-01 isolation 2.3545e-05 (leading 2.3576e-05, rel.dev -1.29e-03), remainder x-(11.12) -1.280e-05 (over isolation 5.44e-01), E_bind-2a = +1.0778e-05 against w11^2/(2a) = 2.3667e-05 u= 0.062500 eps^2/u^2=2.560e+00 isolation 5.9032e-06 (leading 5.9111e-06, rel.dev -1.32e-03), remainder x-(11.12) -1.278e-05 (over isolation 2.16e+00), E_bind-2a = -6.8642e-06 against w11^2/(2a) = 5.9168e-06 u= 0.031250 eps^2/u^2=1.024e+01 isolation 1.4769e-06 (leading 1.4788e-06, rel.dev -1.32e-03), remainder x-(11.12) -1.277e-05 (over isolation 8.65e+00), E_bind-2a = -1.1291e-05 against w11^2/(2a) = 1.4792e-06 u= 0.015625 eps^2/u^2=4.096e+01 isolation 3.6930e-07 (leading 3.6978e-07, rel.dev -1.30e-03), remainder x-(11.12) -1.277e-05 (over isolation 3.46e+01), E_bind-2a = -1.2398e-05 against w11^2/(2a) = 3.6980e-07 u= 0.007812 eps^2/u^2=1.638e+02 isolation 9.2334e-08 (leading 9.2449e-08, rel.dev -1.24e-03), remainder x-(11.12) -1.277e-05 (over isolation 1.38e+02), E_bind-2a = -1.2675e-05 against w11^2/(2a) = 9.2450e-08 reading: (11.30) is sufficient, not necessary. Below the band at P = 0 the remainder stays far below the isolation for every u (the lower edge at P = 0 is exactly M_1+M_2); at P = pi/2 and above the band it tends to a constant of order eps^4 and exceeds the isolation for u of the order of eps === new in v3, I3: tight pair at fixed epsilon (set B_unequal_R, w11 = -u*a), dense diagonalization, L = 120, P = 0 eps'=1.0 u= 2 w11=-0.0242 |w11|/min M = 0.0242 in a band: False weight(r=0,11)=0.8900 E_bind-w11 = 9.315293e-03: rel.dev from the leading form +6.291e-03 (= +0.260 |w11|/M), from the tight form a+a^2/(2w) +2.437e-02; rel.dev E_b''(0) +4.562e-03 (= +0.188 |w11|/M); rel.dev 1- +9.284e-03, +1.328e-02 eps'=1.0 u= 4 w11=-0.0485 |w11|/min M = 0.0485 in a band: False weight(r=0,11)=0.9647 E_bind-w11 = 1.080688e-02: rel.dev from the leading form +1.703e-02 (= +0.351 |w11|/M), from the tight form a+a^2/(2w) +1.887e-02; rel.dev E_b''(0) +1.371e-02 (= +0.283 |w11|/M); rel.dev 1- +3.179e-02, +2.730e-02 eps'=1.0 u= 8 w11=-0.0969 |w11|/min M = 0.0969 in a band: False weight(r=0,11)=0.9863 E_bind-w11 = 1.182203e-02: rel.dev from the leading form +4.038e-02 (= +0.416 |w11|/M), from the tight form a+a^2/(2w) +3.906e-02; rel.dev E_b''(0) +3.300e-02 (= +0.340 |w11|/M); rel.dev 1- +8.288e-02, +5.724e-02 eps'=1.0 u= 16 w11=-0.1939 |w11|/min M = 0.1939 in a band: False weight(r=0,11)=0.9914 E_bind-w11 = 1.280926e-02: rel.dev from the leading form +9.114e-02 (= +0.470 |w11|/M), from the tight form a+a^2/(2w) +8.356e-02; rel.dev E_b''(0) +7.258e-02 (= +0.374 |w11|/M); rel.dev 1- +1.992e-01, +1.216e-01 eps'=1.0 u= 32 w11=-0.3878 |w11|/min M = 0.3878 in a band: False weight(r=0,11)=0.9913 E_bind-w11 = 1.443301e-02: rel.dev from the leading form +2.100e-01 (= +0.542 |w11|/M), from the tight form a+a^2/(2w) +1.735e-01; rel.dev E_b''(0) +1.534e-01 (= +0.396 |w11|/M); rel.dev 1- +4.967e-01, +2.691e-01 eps'=1.0 u= 64 w11=-0.7755 |w11|/min M = 0.7755 in a band: False weight(r=0,11)=0.9862 E_bind-w11 = 1.864784e-02: rel.dev from the leading form +5.510e-01 (= +0.711 |w11|/M), from the tight form a+a^2/(2w) +3.553e-01; rel.dev E_b''(0) +2.896e-01 (= +0.373 |w11|/M); rel.dev 1- +1.547e+00, +6.647e-01 eps'=1.0 u= 128 w11=-1.5511 |w11|/min M = 1.5511 in a band: False weight(r=0,11)=0.9228 E_bind-w11 = 4.408762e-02: rel.dev from the leading form +2.653e+00 (= +1.710 |w11|/M), from the tight form a+a^2/(2w) +7.262e-01; rel.dev E_b''(0) -2.101e+00 (= -1.355 |w11|/M); rel.dev 1- +1.410e+01, +2.168e+00 eps'=0.5 u= 2 w11=-0.0061 |w11|/min M = 0.0061 in a band: False weight(r=0,11)=0.8933 E_bind-w11 = 2.317895e-03: rel.dev from the leading form +1.569e-03 (= +0.259 |w11|/M), from the tight form a+a^2/(2w) +1.978e-02; rel.dev E_b''(0) +1.147e-03 (= +0.189 |w11|/M); rel.dev 1- +2.322e-03, +3.302e-03 eps'=0.5 u= 4 w11=-0.0121 |w11|/min M = 0.0121 in a band: False weight(r=0,11)=0.9688 E_bind-w11 = 2.667675e-03: rel.dev from the leading form +4.216e-03 (= +0.348 |w11|/M), from the tight form a+a^2/(2w) +6.350e-03; rel.dev E_b''(0) +3.438e-03 (= +0.284 |w11|/M); rel.dev 1- +7.853e-03, +6.741e-03 eps'=0.5 u= 8 w11=-0.0242 |w11|/min M = 0.0242 in a band: False weight(r=0,11)=0.9909 E_bind-w11 = 2.868720e-03: rel.dev from the leading form +9.826e-03 (= +0.405 |w11|/M), from the tight form a+a^2/(2w) +9.986e-03; rel.dev E_b''(0) +8.233e-03 (= +0.340 |w11|/M); rel.dev 1- +1.993e-02, +1.391e-02 eps'=0.5 u= 16 w11=-0.0485 |w11|/min M = 0.0485 in a band: False weight(r=0,11)=0.9966 E_bind-w11 = 2.997571e-03: rel.dev from the leading form +2.138e-02 (= +0.441 |w11|/M), from the tight form a+a^2/(2w) +2.096e-02; rel.dev E_b''(0) +1.794e-02 (= +0.370 |w11|/M); rel.dev 1- +4.517e-02, +2.858e-02 eps'=0.5 u= 32 w11=-0.0969 |w11|/min M = 0.0969 in a band: False weight(r=0,11)=0.9980 E_bind-w11 = 3.117421e-03: rel.dev from the leading form +4.538e-02 (= +0.468 |w11|/M), from the tight form a+a^2/(2w) +4.342e-02; rel.dev E_b''(0) +3.759e-02 (= +0.388 |w11|/M); rel.dev 1- +9.876e-02, +5.893e-02 eps'=0.5 u= 64 w11=-0.1939 |w11|/min M = 0.1939 in a band: False weight(r=0,11)=0.9982 E_bind-w11 = 3.297572e-03: rel.dev from the leading form +9.709e-02 (= +0.501 |w11|/M), from the tight form a+a^2/(2w) +8.850e-02; rel.dev E_b''(0) +7.771e-02 (= +0.401 |w11|/M); rel.dev 1- +2.192e-01, +1.238e-01 eps'=0.5 u= 128 w11=-0.3878 |w11|/min M = 0.3878 in a band: False weight(r=0,11)=0.9979 E_bind-w11 = 3.675900e-03: rel.dev from the leading form +2.182e-01 (= +0.563 |w11|/M), from the tight form a+a^2/(2w) +1.791e-01; rel.dev E_b''(0) +1.596e-01 (= +0.412 |w11|/M); rel.dev 1- +5.280e-01, +2.728e-01 reading: at fixed epsilon the relative remainders of the leading forms grow in proportion to |w11|/M; the forms fail for |w11| of order M === new in v3, I2: equal data, on-site binding W0 = w |v> - |22>)/sqrt(2): the flat point of the mixed bands M_1 = M_2 = 1.0, tau_1 = tau_2 = 1.0, w = 1.0; c* = Omega - M = 0.414214; largest weight of v on the flat bands at P = 0 over q: 7.7e-32 P=+0.80: mixed bands [-Delta, Delta], Delta = 0.2971386219; root of 1 + w g_v above them: E = 0.300342915002 (finer q grid: 0.300342915002); E - Delta = 3.2043e-03 = 6.2584e-03 |P|^3; E/|P| = 0.375429; sign changes above: 1, below -Delta: 0 P=+0.40: mixed bands [-Delta, Delta], Delta = 0.1613023588; root of 1 + w g_v above them: E = 0.161712553469 (finer q grid: 0.161712553469); E - Delta = 4.1019e-04 = 6.4093e-03 |P|^3; E/|P| = 0.404281; sign changes above: 1, below -Delta: 0 P=+0.20: mixed bands [-Delta, Delta], Delta = 0.0822915312; root of 1 + w g_v above them: E = 0.082343177123 (finer q grid: 0.082343177123); E - Delta = 5.1646e-05 = 6.4557e-03 |P|^3; E/|P| = 0.411716; sign changes above: 1, below -Delta: 0 P=+0.10: mixed bands [-Delta, Delta], Delta = 0.0413523552; root of 1 + w g_v above them: E = 0.041358823110 (finer q grid: 0.041358823110); E - Delta = 6.4680e-06 = 6.4680e-03 |P|^3; E/|P| = 0.413588; sign changes above: 1, below -Delta: 0 P=+0.05: mixed bands [-Delta, Delta], Delta = 0.0207020497; root of 1 + w g_v above them: E = 0.020702858630 (finer q grid: 0.020702858630); E - Delta = 8.0888e-07 = 6.4711e-03 |P|^3; E/|P| = 0.414057; sign changes above: 1, below -Delta: 0 P=-0.20: mixed bands [-Delta, Delta], Delta = 0.0822915312; root of 1 + w g_v above them: E = 0.082343177123 (finer q grid: 0.082343177123); E - Delta = 5.1646e-05 = 6.4557e-03 |P|^3; E/|P| = 0.411716; sign changes above: 1, below -Delta: 0 dense, L = 240, P=+0.80: eigenvalue 0.300342915002, minus the root -6.1e-16; weight at |r|>L/2 2.6e-08; weight on (r=0, v) 0.0063; neighbours 0.29713619, 2.12475875; eigenvalues in (Delta, 1): 1, in (-1, -Delta): 0 dense, L = 240, P=+0.40: eigenvalue 0.161712553411, minus the root -5.8e-11; weight at |r|>L/2 1.9e-04; weight on (r=0, v) 0.0008; neighbours 0.16130089, 2.03755899; eigenvalues in (Delta, 1): 1, in (-1, -Delta): 0 dense, L = 240, P=0: dimension 1924; eigenvalues with |E| < 1e-9: 962 (2 x number of r: 962), largest of them 7.0e-15; nearest other eigenvalue -1.81616020; root of 1 + w g_v(E,0): -1.81616020; 1 + w g_v(E,0) at E = +0.3, -0.3: 1.0539, 0.9461 reading: for P != 0 one simple eigenvalue lies just above the mixed bands and tends to 0 as c*|P|; at P = 0 the value 0 is the flat point (infinite multiplicity on the infinite chain), isolated from the rest of the spectrum run time: 404 s