1. The problem asks: What do the zeros of an eigenfunction represent? 2. An eigenfunction is a special function associated with an operator, often appearing in physics and mathematics, especially in quantum mechanics and differential equations. 3. The zeros of an eigenfunction are the points where the function's value is exactly zero. 4. These zeros often correspond to nodes or points of no displacement in physical systems, such as standing waves or quantum states. 5. For example, in a vibrating string fixed at both ends, the zeros of the eigenfunction represent points along the string that remain stationary (nodes). 6. In quantum mechanics, zeros of the wavefunction (eigenfunction of the Hamiltonian) indicate positions where the probability density of finding a particle is zero. 7. Thus, zeros of eigenfunctions represent important physical or mathematical boundaries or nodes where the function changes sign or behavior. Final answer: The zeros of an eigenfunction represent points where the function value is zero, often corresponding to nodes or points of no displacement in physical systems, indicating important boundaries or changes in behavior.