To be fair, we're using a linear model (Schrödinger's Equation) but in an infinite dimensional vector space and linear operators on these infinite dimensional v

To be fair, we're using a linear model (Schrödinger's Equation) but in an infinite dimensional vector space and linear operators on these infinite dimensional vector spaces. These aren't your grandfather matrices and there are things in infinite dimension that are rather weird, specifically when it comes to eigenvalues, which QM makes extensive use of. The spectrum of an operator, for example can be a weird mix of discrete values, discrete sets of intervals and/or the entire real line.