you can throw at them. Polk recommends up to 200 watts per channel.  I'd only start there in my consideration of power for the LSi9.

What is your price range and what will your sources be?

I'm not sure why you would need huge amounts of power to drive those speakers. They are rated at 88db/watt, which is pretty efficient. That means 100 watts would give 108 dba of sound which is lethal...lol. 50 watts would give 105 dba, which is again louder than most people would ever go. 100 dba is very very loud.

That is about the same Efficiency as the KEF speakers I have in one of my systems, and I am using a NAD 326BEE there, which is 50w/ch. I have no problem going as loud as I want with that.

I would think that 50-100 watts per channel would do the job nicely, and give them power to spare.

The Cambridge A650 is 75w/ch, and is on sale at Audio Advisor for $500 (marked down form $779), so that is an amp I would buy for them if I was in the market. That amp has had several good reviews, and that is a sweet deal.

The NAD 326BEE is another very good choice, of course, but I suspect that the Cambridge amp might be even better.

With either one you get a damn good amp that will do the job for $500.

sensitivity which is actually drawing 2 watts. Most manufacturers employ the 2.83v value in sensitivity ratings for 4 Ohm speakers so they appear louder per unit of power than they actually are when compared to an 8 Ohm speaker.  In addition, optimistic ratings are the norm. So you need to subtract at least 3dB from your example which might in itself might be debatable. +3dB for two speakers, as much as +6dB for room gain, -6dB for each doubling of distance from 1 meter.

The power dissipation in the amplifier output transistors is significantly higher at 4 Ohms than 8 Ohms, as much a 2.5 times higher, given the same peak current.

In this example, for an 8 ohm load we have : Peak Current = 7.5 Amps, Output Power = 225 Watts, Output Transistor Dissipation = 85 Watts.

And for a 4 ohm load we have : Peak Current = 7.5 Amps, Output Power = 112 Watts, Output Transistor Dissipation = 198 Watts.

Despite keeping the maximum currents the same by halving the power into the 4 ohm load, the power dissipation in the amplifier output transistors is massively increased compared to full output into 8 ohms.  This demonstrates why 4 Ohm speakers are harder on an amplifier than 8 Ohm speakers. Add in sensitivity obfuscation and you really need more amplifier power.

So yes, the Polk LSi9 presents a challenging load for an amplifier.