What Does This Mean: Amplification: 2 x 400 watt per loudspeaker?

Based on a quick read, yes, these are "hybrid" crossover speakers.  Meaning, they have an external crossover that does the crossover between the woofers and the rest of the system.

Each section requires 1 amplifier channel.  There would also be a passive crossover as well, between the tweeter and midrange.

You need 2 channels of amplification per speaker.  One for the woofers and one for the mid/tweet section.

Or it could be there are two 400W amps built in to the speaker and you hook it up to a preamp. I think the specs would've stated 'recommended' amplification  if they wanted you to supply the power.

On their site…

Amplification: A1 or A2 Amplifier, 2 x 400 watt per loudspeaker
Amplification channels: 2 per loudspeaker
Crossover: Woofer/Midrange: Digital crossover stored in processor, 200 Hz. Midrange/Tweeter: Passive crossover, 2200Hz
Connections: Neutrik speakON

Sounds like you need their amp with the active crossover.

Or…

Some other active cross over, and amps feeding SpeakOn connectors.

Your speaker is pretty unique.

The easiest way is to buy a Steinway amp. Check HifiShark.

There was a Lyngdorf A1 for sale on Audiogon last year, but I don't think it sold. Maybe contact the poster?

Bob