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I read a Q of roughly 50 from your waveform which sounds more realistic than Q of 500 in my simulations. Q of 50 gives an acceptable rising envelope, but in no way the voltage boost reported in your posts with a simple LC circuit. Something must be different.
The voltages are straight from the oscilloscope software and the circuit is as per the posted diagram.
I noticed in your simulation that you have one end of the antenna grounded. The return from the antenna in my diagram goes back to the amplifier chip rather than ground because it is a bridge amplifier. Not sure if this will make a difference.
I tried placing a resistor between the antenna and the return to the amplifier. This resulted in lowering the voltage.
According to the documentation on the amplifier, 8 ohms is the required resistance for the output of the amplifier. Reducing the resistance increases the power from the amplifier.
I am in the process of changing the amplifier to increase the power to the antenna.
Will post the final circuit and information when it is complete.