What math should I study if I want to get a basic understanding of quantum mechanics and especially to be able to use the Schrodinger's equation.
A good chunk of the formalism in quantum mechanics involves extensive use of linear algebra. For example, if an operator is given in matrix form, you should know how to compute eigenvalues and eigenvectors. An understanding of group theory is helpful but not required at the basic level. It gives you a better understanding of the formalism for spin, accidental degeneracy in the hydrogen atom, etc. Fourier series and transforms should be at your fingertips. It is very common to frequently switch between momentum space and real space representations. You should familiarize yourself with different representations of the Dirac delta function and learn all the tricks involved in doing Gaussian integrals.
What I described above are tools that one generally uses for a wide variety of problems. Certain mathematical tools, however, might only be used in specific examples. For practical purposes, the Schrödinger equation is solved in the so-called position basis. In this case, your eigenvalue equation is, in general, a partial differential equation (PDE). You should learn how to solve at least the most basic PDEs. You should also look into series solutions of differential equations. Special functions such as Legendre, Laguerre, Airy polynomials etc. would be helpful in certain problems.
Before going inside PDE theory, special functions and all this material other people cited: as for what I understood, you want tu understand Schroedinger equation rather than solve it in strange cases. I'll tell you what I think is the essential to start understanding:
- complex numbers
linear algebra and calculus
complex Hilbert spaces and scalar products
- operators in Hilbert spaces
- Fourier transforms
- Dirac's notation
There are plenty of books about this. Most basic QM books cover at least something about Hilbert spaces, operators and Dirac's notation.