Manuscript Psy02-B@2 · Psy02-B-v2.tex · section · cfd06e2e

Track A4: Gaussian mass energies and stability

Track A4: Gaussian mass energies and stability · lines 779–824

For real Schwartz densities,

Q (f, h) = \iint \frac{f ( x ) h ( y )}{\abs x - y} d^{3} x d^{3} y, \norm f_{C}^{2} = Q (f, f) .

With $f (k) = \int e^{- i k x} f (x) d^{3} x$ ,

Q (f, f) = \frac{1}{( 2 π ) ^{3}} \int \frac{4 π}{\abs k ^{2}} \abs f (k)^{2} d^{3} k .

This form is finite and positive. A wider admissible class can be controlled by Coulomb-energy or Hardy–Littlewood–Sobolev estimates Lieb. Our convention is

E_{G} [f] = GQ (f, f) /2, R [f] = E_{G} [f] /ℏ.

Define

U_{a} (r) = {erf (r / (2 a)) / r, 1/ (π a), r > 0, r = 0.

Fourier transformation of the two Gaussians gives

Q (g_{a} (\cdot - x), g_{a} (\cdot - y)) = U_{a} (\abs x - y) .

Indeed the radial integral uses

\int_{0}^{\infty} e^{- a^{2} k^{2}} \frac{sin ( k r )}{k} d k = \frac{π}{2} erf (r / (2 a)),

obtained by differentiating in $r$ and integrating back from zero.

For equal masses, put

f_{i} = m [g_{a} (\cdot - R_{i} - u /2) - g_{a} (\cdot - R_{i} + u /2)], K_{ij} = GQ (f_{i}, f_{j}) .

Displayed formulas in this passage

unlabeled · line 782
$Q (f, h) = \iint \frac{f ( x ) h ( y )}{\abs x - y} d^{3} x d^{3} y, \norm f_{C}^{2} = Q (f, f) .$
unlabeled · line 788
$Q (f, f) = \frac{1}{( 2 π ) ^{3}} \int \frac{4 π}{\abs k ^{2}} \abs f (k)^{2} d^{3} k .$
unlabeled · line 796
$E_{G} [f] = GQ (f, f) /2, R [f] = E_{G} [f] /ℏ.$
unlabeled · line 801
$U_{a} (r) = {erf (r / (2 a)) / r, 1/ (π a), r > 0, r = 0.$
unlabeled · line 809
$Q (g_{a} (\cdot - x), g_{a} (\cdot - y)) = U_{a} (\abs x - y) .$
unlabeled · line 813
$\int_{0}^{\infty} e^{- a^{2} k^{2}} \frac{sin ( k r )}{k} d k = \frac{π}{2} erf (r / (2 a)),$
unlabeled · line 820
$f_{i} = m [g_{a} (\cdot - R_{i} - u /2) - g_{a} (\cdot - R_{i} + u /2)], K_{ij} = GQ (f_{i}, f_{j}) .$

Track A4: Gaussian mass energies and stability

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