Eigenvalue of the Time Independent Schrodinger Equation 从 is-Programmer.com
There is a statement in [1], say that the following time-independent Schrodinger eqn has discrete spectral if we impose the eigenfunction to be square-integrable over 
. Let me write down the time-independent Schrodinger eqn in a more suggestive form:
I was thinking this problem for some time. My thought was to write 
as exponential function 
. Then 
should satisfies the so-called Riccati eqn. But then I can not figure out how to prove 
should be discrete.
Later on, I found the solution of this problem in a Quantum mechanics text book. Normally, there are two ways for solutions, the first is by power series expansion of , the computation maybe lengthy. The second approach is far more interesting, by factoring the Schrodinger operator into two first order differential operators:
The operator 
are called "ladder operator", since it can transform an eigenfunction with to eigenfunction 
with 
.
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