The Schwartz space of rapidly decreasing functions consists of all infinitely differentiable complex-valued functions on R that vanish at infinity, along with their derivatives, more rapidly than any polynomial. As it is known, an orthonormal basis in the Schwartz space is the so-called Hermite basis, which contains the functions Hn(x)exp(-x2/2), with n=0,1,… and Hn(x) is the n-degree Hermite polynomial. It is also known that the Schwartz space with the L2 norm is a dense subspace of the Banach space L2(R).
Apart from the Hermite basis, are there other bases of the Schwartz space that can be expressed in closed form?
Sure. The reason this one is used is that these functions are also eigenfunctions of the Fourier transform.
But you can do any unitary transformation of this basis and get another basis.
Dear Daniel Grubb, thank you for your answer.
My motive for asking was mainly from physics, particularly from quantum mechanics.
In quantum mechanics, the Hermite basis is also the basis of the energy eigenfunctions of the harmonic oscillator.
The harmonic oscillator is thus an exactly-solvable polynomial potential, and it is the smallest-degree confining polynomial potential.
A confining polynomial potential is an even-degree polynomial potential with a positive leading coefficient, so that it possesses only bound states.
Since the harmonic oscillator is the smallest-degree confining polynomial potential, if there existed other exactly-solvable confining polynomial potentials, their eigenfunctions would also be in Schwartz space, since those potentials would be of higher degree than 2, which is the degree of harmonic oscillator, thus they would be more attractive, and their eigenfunctions would decrease at plus and minus infinity more rapidly than the eigenfunctions of the harmonic oscillator, which are in Schwartz space.
But there are no other exactly-solvable confining polynomial potentials.
So, I wonder whether the other orthonormal bases of Schwartz space could have any physical meaning as eigenfunctions of quantum mechanical potentials.
As you point out the Hermite functions are the energy eigenfunctions for the harmonic oscillator. One aspect here is that the Schrodinger equation for the harmonic oscillator is invariant under the Fourier transform (the second derivatives and x^2 transform into each other).
One way to think about these functions is that they are the result of doing the Graham -Schmidt orthonormalization process starting with the functions 1,x,x^2,x^3, ... The polynomials are dense in the functions that are square integrable with respect to e^{-x^2}dx and so you get an o.n. basis from this process.
If you take *any* sequence of functions whose linear combinations are dense, you can do this same process to get a different o.n. basis. You can even do the polynomials in a different order. Depending on the permutation of the integers used, you may get *very* different results.
For example, do x^0, x^1x^,3, x^2, x^4, x^6, x^5, x^7,x^9, x^11,..
where you take one even exponent, two odd, three even, four odd, etc.
How many solutions of the transformed Hermite’s differential equation which lies in the Schwartz space S of rapidly decreasing functions?
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