Hermite differential equation

For $y^{\prime \prime }-2x{y}^{\prime }+2\lambda y=0$, substitute a power series $y={\sum }_{n\ge 0}{a}_n{x}^n$ to obtain the displayed recurrence relation. It separates into an even solution with $a_0=1,a_1=0$ and an odd solution with $a_0=0,a_1=1$:The ratio of successive terms within each parity tends to zero at every fixed complex $x$, giving infinite radius of convergence. The two initial-value vectors are independent, so these solutions span the solution space. A nonzero polynomial solution exists precisely when $\lambda =N$ is a nonnegative integer: its highest power forces this value, and the recurrence terminates the matching parity. These terminating solutions are nonzero scalar multiples of the Hermite polynomials. Multiplication by $e^{-x^2/2}$ connects them to Hermite oscillator quantization.

For the Hermite differential equation $f^{\prime \prime }-2\xi f^{\prime }+(ϵ-1)f=0$ with real $ϵ$, the wavefunction $e^{-{\xi }^2/2}f$ is square integrable on the real line only if $f$ is a polynomial. To prove this without assuming that even and odd tails cannot cancel, project a square-integrable solution onto its even and odd parts first. Each projection remains square integrable and satisfies the same equation.

In either nonterminating parity sequence, $b_j=a_{2j+p}$ obeysThe tail has constant complex phase. For any fixed $0<\delta <1/2$, comparison with $(1-\delta )/(j+1)$ bounds its coefficient magnitudes below by $K(1-\delta {)}^j/j!$ for sufficiently large $j$. Thus on the positive real axis the tail dominates a positive multiple of ${\xi }^p{e}^{(1-\delta ){\xi }^2}$ after subtraction of finitely many terms. The Gaussian function factor leaves exponential growth $e^{(1/2-\delta ){\xi }^2}$, contradicting the square-integrable condition. Hence each nonzero parity sequence terminates. Its top coefficient forces $ϵ=2n+1$, giving Hermite oscillator quantization and a Hermite polynomial of degree $n$.

A solution of $b^{\prime \prime }+(\Lambda -{\xi }^2)b=0$ decaying at both real infinities has $b=e^{-{\xi }^2/2}{H}_n(\xi )$ with a Hermite polynomial. Substitution gives $H^{\prime \prime }-2\xi H^{\prime }+(\Lambda -1)H=0$; its power-series recurrence terminates at degree $n$ when $\Lambda =2n+1$. The square-integrable Hermite functions give the complete oscillator spectrum, including the ground state $n=0$. This quantization also determines global modes of a quadratically confined Ginzburg-Landau equation after removing drift.

For $-W^{\prime \prime }+2x{W}^{\prime }=EW$, the power series coefficients obey $(n+2)(n+1)a_{n+2}=(2n-E)a_n$. The condition $W(0)=0$ makes every even coefficient zero. A nonzero odd series terminates precisely when $E=2d$ with $d$ an odd positive integer. A polynomial of degree $d$ must have this value of $E$ by comparison of leading coefficients, so these are all possibilities. With $W^{\prime }(0)=1$, the first three polynomial solutions are $x$, $x-2x^3/3$, and $x-4x^3/3+4x^5/15$, with $E=2,6,10$. They are scalar multiples of the odd Hermite polynomials.