Phase boundary

The Minakshisundaram–Pleijel zeta function is a zeta function encoding the eigenvalues of the Laplacian of a compact Riemannian manifold. It was introduced by Template:Harvs. The case of a compact region of the plane was treated earlier by Template:Harvtxt.

Definition

For a compact Riemannian manifold M of dimension N with eigenvalues $\lambda _{1},\lambda _{2},\ldots$ of the Laplace–Beltrami operator Δ the zeta function is given for $\operatorname {Re} (s)$ sufficiently large by

$Z(s)=\operatorname {Tr} (\Delta ^{-s})=\sum _{n=1}^{\infty }\vert \lambda _{n}\vert ^{-s}.$

(where if an eigenvalue is zero it is omitted in the sum). The manifold may have a boundary, in which case one has to prescribe suitable boundary conditions, such as Dirichlet or Neumann boundary conditions.

More generally one can define

$Z(P,Q,s)=\sum _{n=1}^{\infty }{\frac {f_{n}(P)f_{n}(Q)}{\lambda _{n}^{s}}}$

for P and Q on the manifold, where the f_n are normalized eigenfunctions. This can be analytically continued to a meromorphic function of s for all complex s, and is holomorphic for P≠Q.

The only possible poles are simple poles at the points s = N/2, N/2−1, N/2−2,..., 1/2,−1/2, −3/2,... for N odd, and at the points s = N/2, N/2−1, N/2−2, ...,2, 1 for N even. If N is odd then Z(P,P,s) vanishes at s = 0, −1, −2,... If N is even its values can be explicitly by Wiener-Ikehara theorem as a corollary the relation

$\displaystyle Z(P,P,s)\sim {\frac {T^{N/2}}{(2{\sqrt {\pi }})^{N}\Gamma (N/2+1)}}$

where the sign ~ indicates that the quatient of both the sides tend to 1 when T tends to +∞.

The function Z(s) can be recovered from this by integrating Z(P, P, s) over the whole manifold M:

$\displaystyle Z(s)=\int _{M}Z(P,P,s)dP$

Heat kernel

The analytic continuation of the zeta function can be found by expressing it in terms of the heat kernel

$K(P,Q,t)=\sum _{n=1}^{\infty }f_{n}(P)f_{n}(Q)e^{-\lambda _{n}t}$

as the Mellin transform

$Z(P,Q,s)={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }K(P,Q,t)t^{s-1}dt$

In case of heat kernel, for given a Riemannian manifold (M,g) we can take the orthonormal basis of eigenfunctions and obtain the partition function

$Z(s)=\sum _{i=1}^{\infty }e^{-\lambda _{i}s}.$

The poles of the zeta function can be found from the asymptotic behavior of the heat kernel as t→0.

Example

If the manifold is a circle of dimension N=1, then the eigenvalues of the Laplacian are n² for integers n. The zeta function

$Z(s)=\sum _{n\neq 0}{\frac {1}{(n^{2})^{s}}}=\zeta (2s)$

where ζ is the Riemann zeta function.

Applications

Apply the method of heat kernel to asymptotic expansion for Riemannian manifold (M,g) we obtain the two following theorems. Both are the resolutions of the inverse problem in which we get the geometric properties or quantities from spectra of the operators.

1,Minakshisundaram-Pleijel Asymptotic Expansion

Let (M,g) be an n-dimensional Riemannian manifold. The following asymptotic expansions hold as t→0+:

$Z(s)\sim (4\pi s)^{-n/2}\sum _{m=0}^{\infty }a_{m}s^{m}.$

In dim=2, this means that the integral of scalar curvature tells us the Euler characteristic of M, i.e. Gauss-Bonnet Theorem.

In particular,

$a_{0}=Vol(M,g),\ \ \ \ a_{1}={\frac {1}{6}}\int _{M}S(x)dV$

where S(x) is scalar curvature, the trace of the Ricci curvature, on M.

2,Weyl Asymptotic Formula Let M be a compact Riemannian manifold, with eigenvalues $0=\lambda _{0}\leq \lambda _{1}\leq \lambda _{2}\cdots ,$ with each distinct eigenvalue repeated with its multiplicity. Define N(λ) to be the number of eigen values less than or equal to $\lambda$, and let $\omega _{n}$ denote the volume of the unit disk in $\mathbb {R} ^{n}$. Then

$N(\lambda )\sim {\frac {\omega _{n}Vol(M)\lambda ^{n/2}}{(2\pi )^{n}}},$

as λ→∞. Additionally, as k→∞,

$(\lambda _{k})^{n/2}\sim {\frac {(2\pi )^{n}k}{\omega _{n}Vol(M)}}.$

This is also called Weyl's law, refined from Minakshisundaram-Pleijel asymptotic expansion.

References

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• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010