See also perturbative solution of the problem.

Here, the functions H and W depend on two variables, x and y. They include a harmonic part, cubic, and quartic anharmonicity:

H = H_{0} + H_{1} + H_{2}, W = W_{0} + W_{1} + W_{2}.

Harmonic parts are quadratic functions:

H_{0}(x,y) = x^{2}/2 + y^{2}/2, W_{0}(x,y) = w_{x}^{2}(x-x_{0})^{2}/2 + w_{y}^{2}(y-y_{0})^{2}/2.

We call w_{x} and w_{y} "frequencies", and x_{0} and y_{0} "displacements".

Cubic anharmonicity is a cubic polynomial:

H_{1}(x,y) = h_{30}x^{3} + h_{21}x^{2}y + h_{12}xy^{2} + h_{03}y^{3}, W_{1}(x,y) = w_{30}(x-x_{0})^{3} + w_{21}(x-x_{0})^{2}(y-y_{0}) + w_{12}(x-x_{0})(y-y_{0})^{2} + w_{03}(y-y_{0})^{3}.

Quartic anharmonicity is a fourth degree polynomial:

H_{2}(x,y) = h_{40}x^{4} + h_{31}x^{3}y + h_{22}x^{2}y^{2} + h_{13}xy^{3} + h_{04}y^{4}, W_{2}(x,y) = w_{40}(x-x_{0})^{4} + w_{31}(x-x_{0})^{3}(y-y_{0}) + w_{22}(x-x_{0})^{2}(y-y_{0})^{2} + w_{13}(x-x_{0})(y-y_{0})^{3} + w_{04}(y-y_{0})^{4}.

The "energy" is always one, E = 1.

Download Mathematica programs used for this calculation

A paper related to minimizations of quadratic functions

Results of work at BGU with B. Segev

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