Relativistic
The Lorentz transformations are (slightly) more complicated than the Galilean ones, so the solutions to the Schrödinger equation are certainly not Lorentz invariant either, in turn not consistent with special relativity. Also, as shown above in the plausibility argument - the Schrödinger equation was constructed from classical energy conservation rather than the relativistic mass–energy relation
This relativistic equation is Lorentz invariant. The classical equation is not - it is the low-velocity limit of the relativistic equation (velocities much less than the speed of light). This further shows that the Schrödinger equation itself, not just the solutions, is not Lorentz invariant.
Secondly, the equation requires the particles to be the same type, and the number of particles in the system to be constant, since their masses are constants in the equation (kinetic energy terms). This alone means the Schrödinger equation is not compatible with relativity - even the simple equation
allows (in high-energy processes) particles of matter to completely transform into energy by particle-antiparticle annihilation, and enough energy can re-create other particle-antiparticle pairs. So the number of particles and types of particles is not necessarily fixed. For all other intrinsic properties of the particles which may enter the potential function, including mass (such as the harmonic oscillator) and charge (such as electrons in atoms), which will also be constants in the equation, the same problem follows.
Read more about this topic: Schrödinger Equation, Properties, Galilean and Lorentz Transformations
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