My first guess is the higher degree of delocalization of molecular orbitals in cyclic amine in comparison to orbitals in non-cyclic amine.

My first reaction to this question is, as a generalisation, are they? I have a list here that has, for spectra run in CCl4 (so we have a common solvent, diethylamine is listed at 3334, piperidine at 3350 and morpholine at 3351. Incidentally, delocalisation is unlikely to be the answer, because leaving aside the question of whether it can delocalise, in the same solvent pyrrole is 3475 (all in cm-1).

One key feature of such spectra is it is important to make sure, if you are considering fine detail, that all comparisons are carried out under the same conditions. You also have to be careful that the same vibration is considered because the normal vibrations can couple or be modified by what goes on elsewhere, particularly if the phase is changed.

If the changes are real, and not through some other artefact, the force constant changes (same atoms!) usually are dependent on bond strengths - the stronger the bond, the higher the force constant. The N - H bond can also hydrogen bond, which weakens it, and here phase differences, or even concentration differences, might be critical.

I completely agree with Ian. I rapidly checked IR spectra of amines available on the web. For diethyl amine I saw a spectrum with a peak at 3288 cm-1, no mention of the physical state. Another shows 3281 for the N-H stretching "associated" (H bonding N-H...N), and 3410 for the shoulder attributed to the "free" N-H stretch.

The strain due to cyclization may affect the bondi strength and consequently the wavenumber. This effect is observed for cyclic ketones for example, but I don't know the effet on cyclic amines.

In response to Jean-François, aziridine is at 3341 so it is not a significant shift through strain. The effects of strain on adjacent bonds is a bit confusing. As shown in some of my earlier papers (and thanks to my inexperience at the time, maybe not shown as clearly as desired) the external effects are largely due to a polarisation field. As an example, in the three membered ring, four orbitals are deformed towards a given bond, and two away, but the motion of those two and to be more rotational about that given bond axis. With infrared vibrations, the effect depends significantly on the change of electric moment during the vibration, in what direction, and the distance from the ring. With N-H stretch, it appears the various effects more or less cancel. However, when you compare the aziridine stretch with that of pyrrole, it actually reinforces my view that the internal bonds of strained molecules are not delocalised, but merely deformed. But that is another story.