Hermitian Hamiltonians: Matrix versus Schr${ö}$dinger's

We draw attention to the fact that a Hermitian matrix is always diagonalizable and has real discrete spectrum whereas the Hermitian Schr{ö}dinger Hamiltonian: $H=p^2/2μ+V(x)$, may not be so. For instance when $V(x)=x, x^3, -x^2$, $H$ does not have even one real discrete eigenvalue. Textbooks do not highlight this distinction. However, if $H$ has real discrete spectrum, by virtue of the expansion theorem, one can convert the eigenvalue problem $Hψ_n=E_n ψ_n$ into a matrix and get eigenvalues $E_n$ by diagonalizing the matrix. We show, that the thus obtained $E_n$ could be accurate, provided $H$ is devoid of scattering states. We suggest that this could be a simple and apt way to introduce the method of Linear Combination of Atomic Orbitals (LCAO) for finding the spectra of molecules. In textbooks, usually the method of matrix-diagonalization appears meagerly as a degenerate perturbation theory for more than one dimensions.

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