Integral of product of Hermite functions over finite interval
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I am working with the Hermite functions $h_n(x)$ such that $int_{mathbb{R}}h_n(x)h_m(x)e^{-x^2/2}dx=delta_{n,m}.$ So, if $mneq n$, we know that the integral $int_{mathbb{R}}h_n(x)h_m(x)e^{-x^2/2}dx=delta_{n,m}=0$, but I am interested in understanding the behavior of this integral over a finite interval. More precisely if I fix $[-a,a]subset mathbb{R}$, and look at the integral $$I_n:=int_{-sqrt{n}a}^{sqrt{n}b}h_n(x)h_{n+1}(x)e^{-x^2/2}dx,$$
It's clear that $I_nto 0$, but can we say something about the rate at which it goes to 0? I guess that $I_n=O(n^{-2}),$ but I could not find any result in this direction.
One can of course ask for many more generalizations but this is something which I need to use in an estimate I am trying to prove.
Any help shall be highly appreciated.
asymptotics hermite-polynomials
asked Nov 17 at 20:08
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717
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I am working with the Hermite functions $h_n(x)$ such that $int_{mathbb{R}}h_n(x)h_m(x)e^{-x^2/2}dx=delta_{n,m}.$ So, if $mneq n$, we know that the integral $int_{mathbb{R}}h_n(x)h_m(x)e^{-x^2/2}dx=delta_{n,m}=0$, but I am interested in understanding the behavior of this integral over a finite interval. More precisely if I fix $[-a,a]subset mathbb{R}$, and look at the integral $$I_n:=int_{-sqrt{n}a}^{sqrt{n}b}h_n(x)h_{n+1}(x)e^{-x^2/2}dx,$$
It's clear that $I_nto 0$, but can we say something about the rate at which it goes to 0? I guess that $I_n=O(n^{-2}),$ but I could not find any result in this direction.
One can of course ask for many more generalizations but this is something which I need to use in an estimate I am trying to prove.
Any help shall be highly appreciated.
asymptotics hermite-polynomials
asked Nov 17 at 20:08
WhoKnowsWho
717
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I am working with the Hermite functions $h_n(x)$ such that $int_{mathbb{R}}h_n(x)h_m(x)e^{-x^2/2}dx=delta_{n,m}.$ So, if $mneq n$, we know that the integral $int_{mathbb{R}}h_n(x)h_m(x)e^{-x^2/2}dx=delta_{n,m}=0$, but I am interested in understanding the behavior of this integral over a finite interval. More precisely if I fix $[-a,a]subset mathbb{R}$, and look at the integral $$I_n:=int_{-sqrt{n}a}^{sqrt{n}b}h_n(x)h_{n+1}(x)e^{-x^2/2}dx,$$
It's clear that $I_nto 0$, but can we say something about the rate at which it goes to 0? I guess that $I_n=O(n^{-2}),$ but I could not find any result in this direction.
One can of course ask for many more generalizations but this is something which I need to use in an estimate I am trying to prove.
Any help shall be highly appreciated.
asymptotics hermite-polynomials
asked Nov 17 at 20:08
WhoKnowsWho
717
I am working with the Hermite functions $h_n(x)$ such that $int_{mathbb{R}}h_n(x)h_m(x)e^{-x^2/2}dx=delta_{n,m}.$ So, if $mneq n$, we know that the integral $int_{mathbb{R}}h_n(x)h_m(x)e^{-x^2/2}dx=delta_{n,m}=0$, but I am interested in understanding the behavior of this integral over a finite interval. More precisely if I fix $[-a,a]subset mathbb{R}$, and look at the integral $$I_n:=int_{-sqrt{n}a}^{sqrt{n}b}h_n(x)h_{n+1}(x)e^{-x^2/2}dx,$$
It's clear that $I_nto 0$, but can we say something about the rate at which it goes to 0? I guess that $I_n=O(n^{-2}),$ but I could not find any result in this direction.
One can of course ask for many more generalizations but this is something which I need to use in an estimate I am trying to prove.
Any help shall be highly appreciated.
asymptotics hermite-polynomials
asymptotics hermite-polynomials
asked Nov 17 at 20:08
WhoKnowsWho
717
asked Nov 17 at 20:08
WhoKnowsWho
717
asked Nov 17 at 20:08
WhoKnowsWho
717
asked Nov 17 at 20:08
WhoKnowsWho
717
asked Nov 17 at 20:08
WhoKnowsWho
717
717
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