That's the subject of an article here in Norwegian which goes over the (still quite low) possibility of a high-speed rail connection between the two cities. The main reason for the low probability for this is probably the fact that one doesn't even exist between the two largest cities of Oslo and Bergen (304 km) compared to 389 km between Oslo and Trondheim.
A new high-speed rail route between Trondheim and Oslo will cost around 80 billion krone ($14.2 billion), a sum that the community would be unlikely to pay. According to Tore Sandvik in Sør-Trøndelag: "It is unthinkable that such an investment would happen without a route between Bergen and Oslo also being built. That would cost 180 billion ($32 billion). We have a huge investment in railways in the state budget this year, but that is just a fraction of what this line will cost."
The Norwegian parliament has approved a study taking a look at a line between the two large cities, in order to create a basis to make a decision in 2013.
The article also mentions that the line would probably be paid for over 30 years at an interest rate of five percent. In order to get the travel time between the two cities down to below four hours one would need a train with a speed of 160 kph, but the problem here is that the existing rail is a century old with a curve radius (not sure if that's the right term) of 300 metres, but this needs to be increased to 1200 metres to bring a train up to a speed of 160 kph. The high-speed rail envisioned between the two cities would have a speed of 300 kph.
What a waste of money, as existing rail usage is reasonably high you would expect with improvements made to the existing system more peole would use rail. Since Norway has 3rd world highways moeny should be spent on the road network before wasting money on rail.