Rayleigh-Ritz Variational Technique

A technique for computing Eigenfunctions and Eigenvalues. It proceeds by requiring

$$J=\int_a^b [p(x){y_x}^2-q(x)y^2]\,dx$$ (1)

to have a Stationary Value subject to the normalization condition

$$\int_a^b y^2 w(x)\,dx=1$$ (2)

and the boundary conditions

$$py_xy\vert _a^b=0.$$ (3)

This leads to the Sturm-Liouville Equation

$${d\over dx}\left({p{dy\over dx}}\right)+qy+\lambda wy=0,$$ (4)

which gives the stationary values of

$$F[y(x)]={\int_a^b (p{y_x}^2-qy^2)\,dx\over\int_a^b y^2w\,dx}$$ (5)

as

$$F[y_n(x)]=\lambda_n,$$ (6)

where $\lambda_n$ are the Eigenvalues corresponding to the Eigenfunction $y_n$.

References

Arfken, G. ``Rayleigh-Ritz Variational Technique.'' §17.8 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 957-961, 1985.