A polynomial equation is obtained for the solutions of the vibrational frequencies of one-dimensional monoatomic and diatomic lattices with particles connected by identical springs, but with arbitrary springs connecting the end particles to rigid walls. The exact expressions of the different normal modes of oscillations of the linear chain of particles for monoatomic, diatomic and defective lattices are derived in a straightforward way. As special cases of our problem we have considered the effects of different end springs on the vibrational frequencies. One interesting result is that very high frequencies are allowed when the ends of the diatomic lattice are rigidly fixed with the boundary walls.

Polynomial equations are obtained for the solutions of the vibrational frequencies of a simple cubic, primitive orthorhombic and tetragonal Bravais lattices of finite size with particles connected to their nearest neighbours through both central and non-centrarl forces, but with arbitrary forces connecting the surface atoms to the rigid walls. The exact expressions of the different normal modes of oscillations and the amplitudes of vibration of different particles in various modes are obtained by solving three decoupled partial difference equations.

The even and odd parity eigenvalues for the bounded potentialµx² +λx^2m with both positive and negative values ofμ andm=2, 3, 4, 5, 6 are obtained by the method of series solution for any positive value of the coupling constantλ. Whenμ is negative the lower order eigenvalues are closely bunched in pairs in the low-λ regime (λ≪1). The results agree well with existing results.

The unperturbed Hamiltonian of quantum anharmonic oscillator is modified by introducing a simple variational scale parameter. A suitable choice of this parameter makes the eigenvalues rapidly convergent for small size of the determinant in the method of infinite Hill determinant. Simple analytic expressions for the eigenvalues are obtained by matrix diagonalization method.

Exact solutions of the potentialV(r)=−Ze²/(r+β),β>0 are obtained in theN-dimensional space for certain values ofβ by means of factorization of infinite Hill determinant. We discuss some features of the radial wave Schrödinger equation in theN-dimensional space.