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I know what it means that an eigenvalue of a matrix is simple. My question is about the eigenvalues of operators as the Schröndiger one. Here we work with eigenfunctions rather than with eigenvectors so I can not see what the algebraic multiplicity is.

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When you solve the Schrodinger equation, you solve for the possible energies of the system. These are the eigenvalues of the Hamiltonian operator. Just like a single eigenvalue can correspond to multiple eigenvectors, a single energy can correspond to multiple eigenfunctions of the Hamiltonian. This is called degeneracy. An eigenvalue (energy) is called simple if it corresponds to a single eigenfunction of the Hamiltonian.

Take, for example, the one dimensional harmonic oscillator. The Schrodinger equation is - $$\left (-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}+\frac{1}{2}m\omega x^2 \right )\psi=E\psi$$

And the corresponding energies are - $$E_n=\hbar \omega\left (n+\frac{1}{2}\right ), n\in\mathbb N$$

In this case, since each value of $n$ corresponds to a distinct eigenfunction, we know that each energy (eigenvalue) has only one corresponding eigenfunction, and the eigenvalues are thus all simple. This is unlike the case of the $2$ or $3$ dimensional harmonic oscillator, which can have degenerate states - energies with certain multiplicity.

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  • $\begingroup$ So when talking about eigenvalues related to eigenfunction we just need to show that the 'geometric multiplicity' is 1, while when talking about eigenvalues related to eigenvectors we need to show that the algebraic multiplicity is 1? $\endgroup$ – Lucas Pereiro Jul 19 at 8:47
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    $\begingroup$ The terms algebraic/geometric multiplicity relate to eigenvalues of matrices (and linear operators over finite dimensional Hilbert spaces). The reason for the two different terms is related to the fact that eigenvalues and eigenvectors of matrices are usually found using the characteristic polynomial. For operators over infinite dimensional spaces (such as the Hamiltonian), there isn't usually a characteristic polynomial, and people mostly just refer to the 'multiplicity' of an eigenvalue, which is indeed closer to the meaning of the geometric multiplicity in matrices. $\endgroup$ – GSofer Jul 19 at 11:21

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