An eigenfunction is a special type of function that, when operated on by a linear operator, results in a scalar multiple of itself. In other words, the function is only changed by a constant factor when operated on by the linear operator.
The eigenvalue in this equation represents the constant factor by which the eigenfunction is scaled when operated on by the linear operator. In this case, the eigenvalue is x^2-1.
The eigenvalue for this particular problem is determined by solving the differential equation, \frac{d^2}{dx}-x^2: e^{-0.5x^2} = \lambda e^{-0.5x^2}. This results in the characteristic equation \lambda = x^2-1.
The given function is considered an eigenfunction because when operated on by the linear operator, \frac{d^2}{dx}-x^2, it results in a scalar multiple of itself, specifically \lambda e^{-0.5x^2} where \lambda = x^2-1. This satisfies the definition of an eigenfunction.
The term e^{-0.5x^2} is significant because it is the eigenfunction of the linear operator \frac{d^2}{dx}-x^2 with eigenvalue x^2-1. This means that it is a solution to the differential equation and can be used to represent the behavior of the system described by the equation.