10 D-brane backgrounds should play a role in the description of the QCD vacuum analogous to the treatment of the CP $^{N}$− 1 model that we consider in this paper. In both cases, we are dealing with the laminar collapse of a spacetime filling D-brane which occurs locally along a single coordinate direction, with the BCFT for that coordinate acquiring a tachyonic boundary perturbation. For the oscillatory boundary perturbation the spectrum and partition function for the bosonic string have been analyzed both for the case R = R$_{0}$ and R = ∞ in Refs. [11, 13]. Here we are interested in the bulk properties of the zero temperature CP $^{N}$− 1 model, so the case R = ∞ is most directly relevant to our discussion. However, it is instructive to first review the properties of the string spectrum with coordinate compactified at the SU (2) radius R$_{0}$ . The spectrum at R = ∞ can then be understood in terms of Bloch wave states. The R = ∞ theory exhibits a “stringy” band structure, with each band corresponding to one of the eigenstates of the compactified R = R$_{0}$ theory [11, 13]. For the R = ∞ theory with boundary perturbation (16), the Bloch wave basis allows a separation between translational and internal motion analogous to the free string case. Note that the spacing between branes is set by the periodicity of the boundary perturbation (16), 2 πR$_{0}$ . This period is also the circumference of the compactified coordinate at the SU (2) radius, so the theory at the SU (2) compactification point describes a unit cell of the tachyonic crystal in the R = ∞ theory. For nonzero g , the eigenvalue spectrum of the R = ∞ theory consists of bands separated by gaps. Just as in the case of an ordinary particle in a periodic crystal, there are two complementary viewpoints to describe the spectrum of bands and band gaps, the tight binding approximation (TBA) and the nearly-free string (NFS) approximation. In the TBA, the Bloch wave states of the R = ∞ theory are constructed from plane wave superpositions of eigenstates defined on a unit cell. Each eigenstate on the unit cell gives rise to a band of eigenvalues for the R = ∞ theory. The critical boundary perturbation g = 1 2 corresponds to the tight binding limit, in which the states in a band become degenerate, and the spectrum for R = ∞ is just a replication, on each cell, of the discrete spectrum at the SU(2) radius. This is the case when the end of the string is firmly stuck to a single D-brane of the tachyonic crystal and does not hop to neighboring cells, so λ ( X ) = g cos( $_{X/}$√$_{2}$$_{α}$$_{′}$$_{)}$ (16) III. BLOCH WAVES AND BAND STRUCTURE FOR A PROPAGATING STRING: THE CONNECTION BETWEEN R = R 0 AND R = ∞