The coordinate space matrix elements <x|H|x'> represent the expectation value of the Hamiltonian operator for a particular position x. In other words, it tells us the average energy of a particle located at position x in space.
The coordinate space matrix elements <x|H|x'> are used to solve the time-independent Schrödinger equation, which describes the evolution of a quantum system over time. By finding the eigenvalues and eigenvectors of the Hamiltonian operator, we can determine the energy eigenstates of the system and use them to calculate the coordinate space matrix elements.
Yes, the coordinate space matrix elements <x|H|x'> are dependent on the basis set used. This is because the Hamiltonian operator is represented differently in different basis sets, and thus the matrix elements will also be different. However, the expectation value of the Hamiltonian will be the same regardless of the basis set used.
When an external potential is applied, the Hamiltonian operator will change and thus the coordinate space matrix elements <x|H|x'> will also change. This will result in a different expectation value for the energy of the system.
The coordinate space matrix elements <x|H|x'> can be thought of as the probability amplitude for a particle to be found at position x with a particular energy. They represent the quantum mechanical nature of particles, where their position and energy are not sharply defined, but rather described by a probability distribution.