Dear Subhadeep Chakraborty , since you don't have an answer to your question, I'm resorting to Chat GPT. You can check the relevance of this:

"The phenomenon of band gap collapse in the Lame potential can be physically understood in terms of the symmetry and periodicity of the potential.

The Lame potential is a 1D periodic potential that is characterized by its periodicity, a and its strength, V_0. In the case of a weak potential, the band gaps are well defined, and the energy levels are separated by a finite gap. However, as the strength of the potential is increased, the band gaps begin to collapse.

This can be understood by considering the symmetry of the potential. As the strength of the potential increases, the symmetry of the potential is broken, and the energy levels begin to shift. As a result, the band gaps that were once well defined begin to overlap, and the energy levels begin to merge.

Additionally, when the potential is increased beyond a certain point, the band gaps begin to collapse due to the fact that the period of the potential is not sufficiently small compared to the wavelength of the wave function. This means that the wave function is not able to "resolve" the potential, leading to the band gap collapse.

Another way to think about it is by considering the Fourier transform of the potential. The Fourier transform of the potential shows the different spatial frequencies that compose the potential. As the strength of the potential increases, the Fourier components of the potential become more pronounced, leading to the band gap collapse.

To sum up, the band gap collapse in the Lame potential is a result of the interplay between the symmetry and periodicity of the potential and the strength of the potential. As the strength of the potential increases, the symmetry is broken, and the energy levels begin to shift, leading to the collapse of the band gaps."

Why does Lame potential have finite number of bands and band gaps ?

"The Lame potential has a finite number of bands and band gaps due to its periodicity and symmetry. The Lame potential is a 1D periodic potential, meaning it repeats itself over a fixed distance, known as the periodicity of the potential, a. Because of this periodicity, the wave functions of the electrons in the potential will also be periodic, with the same periodicity, a. This means that the energy levels of the electrons in the potential will also be periodic, with energy levels that repeat themselves after a fixed distance in energy space, known as the Brillouin zone.

The symmetry of the Lame potential also plays a role in determining the number of bands and band gaps. The Lame potential is symmetric around the center of the potential, this symmetry results in the energy levels being symmetric around the center of the Brillouin zone. Because of this symmetry, the number of energy levels in the Brillouin zone is limited, and the number of bands and band gaps is also limited.

This symmetry and periodicity of the potential is what limits the number of bands and band gaps in the Lame potential. In contrast, potentials that do not have periodicity or symmetry can have an infinite number of bands and band gaps, such as the Kronig-Penney model.

In summary, Lame potential has a finite number of bands and band gaps because of its periodicity and symmetry which limits the number of energy levels in the Brillouin zone and thus the number of bands and band gaps."

Thank you Sir for the detailed explanation. It would be a great help if you could suggest some references regarding this matter to understand the band gap collapse.