Your restrictions don't make a lot of sense to me, so here are some comments. In order for an antenna to make suitable connection to the 2000 ohm input of your tuner, the antenna terminals must connect directly to your tuner terminals, with no transmission line in between. Also, since you insist on using a single wire, we must know what counterpoise is to be provided by the tuner, or by the environment to work with this single wire. I don't think it is possible to couple a receiver that is very small (relative to a wavelength) to the end of a wire effectively without knowing what to do with the "other" terminal of the receiver, the terminal that provides the voltage reference. This applies in free space. Near the ground it is a different matter. So, where is the receiver?
If the receiver does indeed have a voltage reference, by virtue of being grounded for example, then the answer is that your antenna should be one half wavelength or any integer multiple of one half wavelength. The voltage distribution on a half wavelength wire antenna has a current maxima in the center, along with a voltage minima at the same point. At the ends, the opposite occurs, you have a current minima and voltage maxima. If your receiver has infinite input impedance then this works out fine. However, 2000 ohms is not infinite, so a small adjustment in length is needed to find the 2000 ohm point on the wire rather than the maxima. This change in length can be slightly shorter or slightly longer than half wavelength, and it won't be very much.