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How do we show that ##\psi_n(x)## has ##(n-1)## zeros for all ##n\in Z^+##?
Assuming ##\psi_k(x)## has ##(k-1)## zeros for some ##k\in Z^+##, by oscillation theorem, we can only get ##\psi_{k+1}(x)## has ##\geq k## zeros.
Also, how do we show that ##\psi_1(x)##, the eigenfunction corresponding to ##E_1##, has no zero? Oscillation theorem may permit ##\psi_1(x)## to have 3 zeros, ##\psi_2(x)## to have 5 zeros, ##\psi_3(x)## to have 20 zeros, etc.