Eigenvalue Problems

1. Initial Value Problems.
2. Boundary Value Problems
3. Eigenvalue (characteristic-value) Problems Even if the boundary conditions of the differential equation are available, the solutions can only exist for some specific values of a parameter in the system. For example,
   $$-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2}+V(x)\psi(x)=E\psi(x)$$
   • In Schrödinger equation, there are solutions that $\psi(x)$ goes to zero at infinity for certain values of the energy E.
   • These E$_n$ values satisfying this condition are the eigenvalues of the differential equation.

• See the following 2nd degree differential equation:

  $$y''=f(x,y,y':\lambda)$$

• Again, let the boundary conditions of this equation to be given at both ends.
• If these conditions can only be satisfied for certain $\lambda$ values, we call it the eigenvalue problem.
  • e.g.: Vibrations of a wire with both ends fixed give stable solution only for certain wavelengths.
  • e.g.: Solutions of the Schrödinger equation that are zero at infinity exist only for certain energy eigenvalues.
• Some boundary value problems in physics/engineering have a solution based on an eigenvalue.
• In terms of numerical solution, boundary value and eigenvalue problems are solved by the same method.

1

For example, standing waves are the solutions of the following differential equation for a string fixed at both ends: $$\frac{d^2y(x)}{dx^2}+k^2y(x)=0$$ Here $k=2\pi/\lambda$ represents the wavenumber. If the two ends of the L-length string are fixed, then corresponding boundary values are: $$y(0)=y(L)=0$$ Although there are sufficient boundary conditions, there are only solutions for certain $k$ values. It is impossible to satisfy these boundary conditions for other $k$ values. The standing wave solution in the string: $$y(x)=Asinkx+Bcoskx$$

Boundary conditions to find the coefficients A, B;

According to these results, there is only one set of solution as $k=\pi/L, 2\pi/L, \ldots$ values. ( $k_n=n\pi/L \rightarrow$ eigenvalue(s).)2 Another example is the Schrödinger equation in quantum mechanics. If we write it in one-dimensional space,

$$-\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2}+V(x)\psi(x)=E\psi(x)$$

Here $V(x)$ is the potential energy function of the particle, and $E$ is its total energy. Boundary conditions for the wave function are given for $x= \pm \infty$:

$$\psi(\pm \infty) \rightarrow 0$$

Again, this differential equation has solutions satisfying the boundary conditions only for certain $E$ eigenvalues. For eigenvalue problems, the trial-and-error (shooting) method is also used.

• However, an estimated value is given to the eigenvalue instead of giving to the derivatives at the boundary.
• Then, a trial solution is obtained.
• By comparing this trial solution with the value at the boundary condition, the eigenvalue is readjusted and another trial is performed.
• Finally, the solution is to be found when the true eigenvalue is approached within a certain margin of error.