Dispersion relations with crossing symmetry for<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>π</mml:mi><mml:mi>π</mml:mi></mml:math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>D</mml:mi></mml:math>- and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>F</mml:mi></mml:math>-wave amplitudes

A set of once subtracted dispersion relations with imposed crossing symmetry condition for the $\ensuremath{\pi}\ensuremath{\pi}$ $D$- and $F$-wave amplitudes is derived and analyzed. An example of numerical calculations in the effective two-pion mass range from the threshold to 1.1 GeV is presented. It is shown that these new dispersion relations impose quite strong constraints on the analyzed $\ensuremath{\pi}\ensuremath{\pi}$ interactions and are very useful tools to test the $\ensuremath{\pi}\ensuremath{\pi}$ amplitudes. One of the goals of this work is to provide a complete set of equations required for easy use. Full analytical expressions are presented. Along with the well-known dispersion relations successful in testing the $\ensuremath{\pi}\ensuremath{\pi}$ $S$- and $P$-wave amplitudes, those presented here for the $D$ and $F$ waves give a complete set of tools for analyses of the $\ensuremath{\pi}\ensuremath{\pi}$ interactions.

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