Asymptotic expansion of heat operator $e^{-Delta{t}}$ and $e^{-mathcal{D}t}$ of Dirac operator
$begingroup$
For a closed Riemannian manifold $M$ of $n$-dimension, we consider the Laplace-Beltrami operator $Delta$.
It is known that we have an asymptotic expansion for the trace of heat operator $e^{-Delta{t}}$ as follows
$$
mathrm{tr}(e^{-Delta{t}})=sum_{lambda}e^{-lambda{t}}overset{tdownarrow0}{sim}t^{-frac{n}{2}}sum_{n} alpha_{n}t^{n},
$$
where $lambda$ runs over the set of spectrum of Laplacian $Delta$.
My question is that
Denote by $mathcal{D}$ the Dirac operator whose square coincides with the Laplacian, $i.e.$ $mathcal{D}^{2}=Delta$.
Then the sum of positive eigenvalues of an operator $e^{-mathcal{D}t}$ $$sum_{lambdainmathrm{Sp}(Delta)}e^{-sqrt{lambda}{t}}$$ has an asymptotic expansion around $t=0$?
If it exists, then is it possible to induce a relation between coefficients?
I know that the proof for the case of heat operator follows from the construction of heat kernel. But I wonder that the same construction can be applied to the Dirac operator.
Thank you for your time and effort.
differential-geometry pde asymptotics spectral-theory differential-operators
$endgroup$
add a comment |
$begingroup$
For a closed Riemannian manifold $M$ of $n$-dimension, we consider the Laplace-Beltrami operator $Delta$.
It is known that we have an asymptotic expansion for the trace of heat operator $e^{-Delta{t}}$ as follows
$$
mathrm{tr}(e^{-Delta{t}})=sum_{lambda}e^{-lambda{t}}overset{tdownarrow0}{sim}t^{-frac{n}{2}}sum_{n} alpha_{n}t^{n},
$$
where $lambda$ runs over the set of spectrum of Laplacian $Delta$.
My question is that
Denote by $mathcal{D}$ the Dirac operator whose square coincides with the Laplacian, $i.e.$ $mathcal{D}^{2}=Delta$.
Then the sum of positive eigenvalues of an operator $e^{-mathcal{D}t}$ $$sum_{lambdainmathrm{Sp}(Delta)}e^{-sqrt{lambda}{t}}$$ has an asymptotic expansion around $t=0$?
If it exists, then is it possible to induce a relation between coefficients?
I know that the proof for the case of heat operator follows from the construction of heat kernel. But I wonder that the same construction can be applied to the Dirac operator.
Thank you for your time and effort.
differential-geometry pde asymptotics spectral-theory differential-operators
$endgroup$
add a comment |
$begingroup$
For a closed Riemannian manifold $M$ of $n$-dimension, we consider the Laplace-Beltrami operator $Delta$.
It is known that we have an asymptotic expansion for the trace of heat operator $e^{-Delta{t}}$ as follows
$$
mathrm{tr}(e^{-Delta{t}})=sum_{lambda}e^{-lambda{t}}overset{tdownarrow0}{sim}t^{-frac{n}{2}}sum_{n} alpha_{n}t^{n},
$$
where $lambda$ runs over the set of spectrum of Laplacian $Delta$.
My question is that
Denote by $mathcal{D}$ the Dirac operator whose square coincides with the Laplacian, $i.e.$ $mathcal{D}^{2}=Delta$.
Then the sum of positive eigenvalues of an operator $e^{-mathcal{D}t}$ $$sum_{lambdainmathrm{Sp}(Delta)}e^{-sqrt{lambda}{t}}$$ has an asymptotic expansion around $t=0$?
If it exists, then is it possible to induce a relation between coefficients?
I know that the proof for the case of heat operator follows from the construction of heat kernel. But I wonder that the same construction can be applied to the Dirac operator.
Thank you for your time and effort.
differential-geometry pde asymptotics spectral-theory differential-operators
$endgroup$
For a closed Riemannian manifold $M$ of $n$-dimension, we consider the Laplace-Beltrami operator $Delta$.
It is known that we have an asymptotic expansion for the trace of heat operator $e^{-Delta{t}}$ as follows
$$
mathrm{tr}(e^{-Delta{t}})=sum_{lambda}e^{-lambda{t}}overset{tdownarrow0}{sim}t^{-frac{n}{2}}sum_{n} alpha_{n}t^{n},
$$
where $lambda$ runs over the set of spectrum of Laplacian $Delta$.
My question is that
Denote by $mathcal{D}$ the Dirac operator whose square coincides with the Laplacian, $i.e.$ $mathcal{D}^{2}=Delta$.
Then the sum of positive eigenvalues of an operator $e^{-mathcal{D}t}$ $$sum_{lambdainmathrm{Sp}(Delta)}e^{-sqrt{lambda}{t}}$$ has an asymptotic expansion around $t=0$?
If it exists, then is it possible to induce a relation between coefficients?
I know that the proof for the case of heat operator follows from the construction of heat kernel. But I wonder that the same construction can be applied to the Dirac operator.
Thank you for your time and effort.
differential-geometry pde asymptotics spectral-theory differential-operators
differential-geometry pde asymptotics spectral-theory differential-operators
asked Nov 30 '18 at 11:54
Junhyeong KimJunhyeong Kim
558
558
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
We define the spectral zeta function of the Laplacian $Delta$ by
$$
zeta_{Delta}(s) = sum_{lambdainmathrm{Sp}(Delta)}lambda^{-s}.
$$
It is known that $zeta_{Delta}(s)$ converges absolutely over $mathrm{Re}(s)>frac{n}{2}$, and has an analytic continuation to a meromorphic function over all complex plane.
It can be expressed by
$$zeta_{Delta}(s) = frac{1}{Gamma(s)}int_{0}^{infty}t^{s-1}sum_{lambdainmathrm{Sp}(Delta)}e^{-lambda{t}}dt,$$
where $Gamma(s):=int_{0}^{infty}t^{s-1}e^{-t}dt$ is the Gamma function.
From the identities
$$
Gamma(s)lambda^{-s}=int_{0}^{infty}t^{s-1}e^{-lambda{t}}dt
$$
and
$$
Gamma(2s)lambda^{-s}=int_{0}^{infty}t^{2s-1}e^{-sqrt{lambda}t}dt,
$$
we deduce the following relation
$$
zeta_{Delta}(s) = frac{1}{Gamma(s)}int_{0}^{infty}t^{s-1}sum_{lambdainmathrm{Sp}(Delta)}e^{-lambda{t}}dt \
=frac{1}{Gamma(2s)}int_{0}^{infty}t^{2s-1}sum_{lambdainmathrm{Sp}(Delta)}e^{-sqrt{lambda}t}dt.
$$
Since the spectral zeta function $zeta_{Delta}(s)$ is meromorphic over a complex plane, $i.e.$ is analytic except for discrete singularities, the series
$$
sum_{lambdainmathrm{Sp}(Delta)}e^{-sqrt{lambda}t}
$$
has an asymptotic expansion.
$endgroup$
$begingroup$
If the spectral zeta function is meromorphic over a complex plane and $zeta_Delta(s) Gamma(2s)$ satisfies the conditions of some tauberian theorem then the series has an asymptotic expansion given by the poles of $zeta_Delta(s) Gamma(2s)$. (try with $-1 + sum_n e^{-n^2 x}$ whose Mellin transform $Gamma(s) zeta(2s)$ has only one pole)
$endgroup$
– reuns
Dec 8 '18 at 20:24
add a comment |
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1 Answer
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active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
We define the spectral zeta function of the Laplacian $Delta$ by
$$
zeta_{Delta}(s) = sum_{lambdainmathrm{Sp}(Delta)}lambda^{-s}.
$$
It is known that $zeta_{Delta}(s)$ converges absolutely over $mathrm{Re}(s)>frac{n}{2}$, and has an analytic continuation to a meromorphic function over all complex plane.
It can be expressed by
$$zeta_{Delta}(s) = frac{1}{Gamma(s)}int_{0}^{infty}t^{s-1}sum_{lambdainmathrm{Sp}(Delta)}e^{-lambda{t}}dt,$$
where $Gamma(s):=int_{0}^{infty}t^{s-1}e^{-t}dt$ is the Gamma function.
From the identities
$$
Gamma(s)lambda^{-s}=int_{0}^{infty}t^{s-1}e^{-lambda{t}}dt
$$
and
$$
Gamma(2s)lambda^{-s}=int_{0}^{infty}t^{2s-1}e^{-sqrt{lambda}t}dt,
$$
we deduce the following relation
$$
zeta_{Delta}(s) = frac{1}{Gamma(s)}int_{0}^{infty}t^{s-1}sum_{lambdainmathrm{Sp}(Delta)}e^{-lambda{t}}dt \
=frac{1}{Gamma(2s)}int_{0}^{infty}t^{2s-1}sum_{lambdainmathrm{Sp}(Delta)}e^{-sqrt{lambda}t}dt.
$$
Since the spectral zeta function $zeta_{Delta}(s)$ is meromorphic over a complex plane, $i.e.$ is analytic except for discrete singularities, the series
$$
sum_{lambdainmathrm{Sp}(Delta)}e^{-sqrt{lambda}t}
$$
has an asymptotic expansion.
$endgroup$
$begingroup$
If the spectral zeta function is meromorphic over a complex plane and $zeta_Delta(s) Gamma(2s)$ satisfies the conditions of some tauberian theorem then the series has an asymptotic expansion given by the poles of $zeta_Delta(s) Gamma(2s)$. (try with $-1 + sum_n e^{-n^2 x}$ whose Mellin transform $Gamma(s) zeta(2s)$ has only one pole)
$endgroup$
– reuns
Dec 8 '18 at 20:24
add a comment |
$begingroup$
We define the spectral zeta function of the Laplacian $Delta$ by
$$
zeta_{Delta}(s) = sum_{lambdainmathrm{Sp}(Delta)}lambda^{-s}.
$$
It is known that $zeta_{Delta}(s)$ converges absolutely over $mathrm{Re}(s)>frac{n}{2}$, and has an analytic continuation to a meromorphic function over all complex plane.
It can be expressed by
$$zeta_{Delta}(s) = frac{1}{Gamma(s)}int_{0}^{infty}t^{s-1}sum_{lambdainmathrm{Sp}(Delta)}e^{-lambda{t}}dt,$$
where $Gamma(s):=int_{0}^{infty}t^{s-1}e^{-t}dt$ is the Gamma function.
From the identities
$$
Gamma(s)lambda^{-s}=int_{0}^{infty}t^{s-1}e^{-lambda{t}}dt
$$
and
$$
Gamma(2s)lambda^{-s}=int_{0}^{infty}t^{2s-1}e^{-sqrt{lambda}t}dt,
$$
we deduce the following relation
$$
zeta_{Delta}(s) = frac{1}{Gamma(s)}int_{0}^{infty}t^{s-1}sum_{lambdainmathrm{Sp}(Delta)}e^{-lambda{t}}dt \
=frac{1}{Gamma(2s)}int_{0}^{infty}t^{2s-1}sum_{lambdainmathrm{Sp}(Delta)}e^{-sqrt{lambda}t}dt.
$$
Since the spectral zeta function $zeta_{Delta}(s)$ is meromorphic over a complex plane, $i.e.$ is analytic except for discrete singularities, the series
$$
sum_{lambdainmathrm{Sp}(Delta)}e^{-sqrt{lambda}t}
$$
has an asymptotic expansion.
$endgroup$
$begingroup$
If the spectral zeta function is meromorphic over a complex plane and $zeta_Delta(s) Gamma(2s)$ satisfies the conditions of some tauberian theorem then the series has an asymptotic expansion given by the poles of $zeta_Delta(s) Gamma(2s)$. (try with $-1 + sum_n e^{-n^2 x}$ whose Mellin transform $Gamma(s) zeta(2s)$ has only one pole)
$endgroup$
– reuns
Dec 8 '18 at 20:24
add a comment |
$begingroup$
We define the spectral zeta function of the Laplacian $Delta$ by
$$
zeta_{Delta}(s) = sum_{lambdainmathrm{Sp}(Delta)}lambda^{-s}.
$$
It is known that $zeta_{Delta}(s)$ converges absolutely over $mathrm{Re}(s)>frac{n}{2}$, and has an analytic continuation to a meromorphic function over all complex plane.
It can be expressed by
$$zeta_{Delta}(s) = frac{1}{Gamma(s)}int_{0}^{infty}t^{s-1}sum_{lambdainmathrm{Sp}(Delta)}e^{-lambda{t}}dt,$$
where $Gamma(s):=int_{0}^{infty}t^{s-1}e^{-t}dt$ is the Gamma function.
From the identities
$$
Gamma(s)lambda^{-s}=int_{0}^{infty}t^{s-1}e^{-lambda{t}}dt
$$
and
$$
Gamma(2s)lambda^{-s}=int_{0}^{infty}t^{2s-1}e^{-sqrt{lambda}t}dt,
$$
we deduce the following relation
$$
zeta_{Delta}(s) = frac{1}{Gamma(s)}int_{0}^{infty}t^{s-1}sum_{lambdainmathrm{Sp}(Delta)}e^{-lambda{t}}dt \
=frac{1}{Gamma(2s)}int_{0}^{infty}t^{2s-1}sum_{lambdainmathrm{Sp}(Delta)}e^{-sqrt{lambda}t}dt.
$$
Since the spectral zeta function $zeta_{Delta}(s)$ is meromorphic over a complex plane, $i.e.$ is analytic except for discrete singularities, the series
$$
sum_{lambdainmathrm{Sp}(Delta)}e^{-sqrt{lambda}t}
$$
has an asymptotic expansion.
$endgroup$
We define the spectral zeta function of the Laplacian $Delta$ by
$$
zeta_{Delta}(s) = sum_{lambdainmathrm{Sp}(Delta)}lambda^{-s}.
$$
It is known that $zeta_{Delta}(s)$ converges absolutely over $mathrm{Re}(s)>frac{n}{2}$, and has an analytic continuation to a meromorphic function over all complex plane.
It can be expressed by
$$zeta_{Delta}(s) = frac{1}{Gamma(s)}int_{0}^{infty}t^{s-1}sum_{lambdainmathrm{Sp}(Delta)}e^{-lambda{t}}dt,$$
where $Gamma(s):=int_{0}^{infty}t^{s-1}e^{-t}dt$ is the Gamma function.
From the identities
$$
Gamma(s)lambda^{-s}=int_{0}^{infty}t^{s-1}e^{-lambda{t}}dt
$$
and
$$
Gamma(2s)lambda^{-s}=int_{0}^{infty}t^{2s-1}e^{-sqrt{lambda}t}dt,
$$
we deduce the following relation
$$
zeta_{Delta}(s) = frac{1}{Gamma(s)}int_{0}^{infty}t^{s-1}sum_{lambdainmathrm{Sp}(Delta)}e^{-lambda{t}}dt \
=frac{1}{Gamma(2s)}int_{0}^{infty}t^{2s-1}sum_{lambdainmathrm{Sp}(Delta)}e^{-sqrt{lambda}t}dt.
$$
Since the spectral zeta function $zeta_{Delta}(s)$ is meromorphic over a complex plane, $i.e.$ is analytic except for discrete singularities, the series
$$
sum_{lambdainmathrm{Sp}(Delta)}e^{-sqrt{lambda}t}
$$
has an asymptotic expansion.
answered Dec 7 '18 at 13:03
Junhyeong KimJunhyeong Kim
558
558
$begingroup$
If the spectral zeta function is meromorphic over a complex plane and $zeta_Delta(s) Gamma(2s)$ satisfies the conditions of some tauberian theorem then the series has an asymptotic expansion given by the poles of $zeta_Delta(s) Gamma(2s)$. (try with $-1 + sum_n e^{-n^2 x}$ whose Mellin transform $Gamma(s) zeta(2s)$ has only one pole)
$endgroup$
– reuns
Dec 8 '18 at 20:24
add a comment |
$begingroup$
If the spectral zeta function is meromorphic over a complex plane and $zeta_Delta(s) Gamma(2s)$ satisfies the conditions of some tauberian theorem then the series has an asymptotic expansion given by the poles of $zeta_Delta(s) Gamma(2s)$. (try with $-1 + sum_n e^{-n^2 x}$ whose Mellin transform $Gamma(s) zeta(2s)$ has only one pole)
$endgroup$
– reuns
Dec 8 '18 at 20:24
$begingroup$
If the spectral zeta function is meromorphic over a complex plane and $zeta_Delta(s) Gamma(2s)$ satisfies the conditions of some tauberian theorem then the series has an asymptotic expansion given by the poles of $zeta_Delta(s) Gamma(2s)$. (try with $-1 + sum_n e^{-n^2 x}$ whose Mellin transform $Gamma(s) zeta(2s)$ has only one pole)
$endgroup$
– reuns
Dec 8 '18 at 20:24
$begingroup$
If the spectral zeta function is meromorphic over a complex plane and $zeta_Delta(s) Gamma(2s)$ satisfies the conditions of some tauberian theorem then the series has an asymptotic expansion given by the poles of $zeta_Delta(s) Gamma(2s)$. (try with $-1 + sum_n e^{-n^2 x}$ whose Mellin transform $Gamma(s) zeta(2s)$ has only one pole)
$endgroup$
– reuns
Dec 8 '18 at 20:24
add a comment |
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