Great Country Academician

After chatting with Zhou Hai about the Weyl-Berry conjecture in the classroom, Xu Chuan locked himself in the library again.

It has to be said that although the Weyl-Berry conjecture is a world-class conjecture, and the difficulty can even be ranked around T3, there is really not much information about this conjecture.

However, with the research, Xu Chuan unexpectedly discovered that the first asymptotic theorem of the Weyl conjecture, the predecessor of the Weyl-Berry conjecture, is actually the same as the Sommerfeld quantization condition in early quantum mechanics.

This further stimulated his interest in the Weyl-Berry conjecture.

Sure enough, mathematics and physics complement each other!

For more than a month, Xu Chuan learned about the Weyl-Berry conjecture in the library.

Starting from the elliptic operator, to the differential operator and then to the Laplacian operator, Xu Chuan has not let go of every basic book related to the Weyl-Berry conjecture.

In the library, Xu Chuan closed the book in his hand, then took out his laptop from his bag, created a new document, and wrote:

[Proof of Spectral Asymptotics and Weyl_Berry Conjecture on Connected Regions with Fractal Boundaries! 】

A long period of study, coupled with the mathematical knowledge brought back by his rebirth, gave him a deep enough understanding of the spectral asymptotics on connected regions with fractal boundaries.

Although it is impossible to directly prove the Weyl_Berry conjecture, after weakening the Weyl_Berry conjecture, the connected fractal drum that meets the "cut" condition is a kind of natural connected fractal drum. Xu Chuan thinks he can have a try.

At least in this area, he already has some ideas in his mind, and he can write them out regardless of whether he succeeds or not.

[Introduction: In 1993, Lapidi and Pomerans proved that the one-dimensional Weyl-Berry conjecture is true, but for the high-dimensional Weyl-Berry conjecture, the situation becomes very complicated. The high-dimensional Weyl-Berry conjecture In the Minkowski framework generally no longer holds. 】

[But at the same time, two mathematicians, Levitin M and Vasiliev, proved that under a special kind of high-dimensional example, the Weyl-Berry conjecture is established under the Minkowski framework. 】

[All this shows that using the Minkowski framework cannot fully cover all the complexity of the problem, so the correct formulation of the Weyl-Berry conjecture should be:

"Is there a certain fractal framework, so that the boundary Ω is measurable under this fractal framework, and the Weyl-Berry conjecture is established under this fractal framework?"】

After writing down the title and introduction, Xu Chuan skipped the text and typed a few blank lines.

Citation:

[[1] Kigami J, Lapidus M L. Weyl on the problem of spectral distribution of Laplacian operators, P. C. F. Self-similar sets. Acta Mathematics and Physics, 1993, 158: 93-125]

[[2] Spectral asymptotics, update theorem and Berry conjecture for a class of fractals. Journal of Mathematics and Engineering, 1996, 72(3): 188-214]

【.】

There are not many references, less than a slap in the face.

It can only be said that not many people have made much contribution in this area.

In fact, this is exactly the case. Since 1979, M. V. Berry, a physicist in the country where the sun never sets, extended the Weyl conjecture to the case where Ω is a fractal region when studying the scattering of light waves on fractal objects. For decades, Numerous mathematicians, mathematics enthusiasts, and physicists have worked hard on the spectral asymptotic region on the connected region with fractal boundaries.

However, thirty years have passed, and apart from the fact that in 1993, two mathematicians Lapidi and Pomeranz proved that the one-dimensional Weyl-Berry conjecture is true, there are almost no new achievements.

Countless mathematicians, mathematics enthusiasts and physicists have worked hard for more than 30 years, but no one has successfully turned the Weyl-Berry conjecture into the Weyl-Berry theorem.

But here lies the charm of mathematics and physics. The conjectures are like heavy fruits hanging on the tree. Both mathematicians and physicists can see the attractive bright red and full fruit shape.

All that is waiting is for a mathematician or physicist to build a ladder and climb up to pick it up.

Well, Uncle Newton is an exception. Others use a ladder to climb up to pick it, but he falls the apple on his head.

After typing down the title and introduction, Xu Chuan put the computer away, took out a stack of A4 manuscript paper from his school bag, and began to continue writing the thoughts in his heart.

The library of Nantah University is very large, and some areas are quite quiet.

Just like where he is now, because the books stored are relatively remote books, there are not many people around, so Xu Chuan ran back to the dormitory lazily.

Assuming Ω Rn is a bounded open set, we consider the following eigenvalue problem of Dirichlet-Laplace operator: (P){-△ u=λu, x∈Ω; u|Ω= 0

Then the problem (P) has a discrete spectrum {λi}i∈N, and can be arranged in a row: 0 \u003cλ1≤λ2 ≤λk≤. . . . .

Here limk→+∞λk =+∞, we are interested in which geometric quantities of Ω are spectrally invariant (that is to say, uniquely determined by the spectrum {λi}i∈N).

This aspect of the problem relies on studying the asymptotic behavior of the eigenvalue λk as k→+∞. For λ \u003e 0, define.

The black signature pen in his hand continuously outlines symbols and words on the white manuscript paper.

For Xu Chuan, he has already ignored everything around him after entering the proof process. Everything in the world no longer exists in his eyes, only the manuscript paper and pen on the table, and the lines of calculations and formulas output from his mind. Word.

When numbers and theorems, formulas and symbols dance under the tip of the pen, the beauty brought by the perfect rhythm keeps emerging in Xu Chuan's mind, making him intoxicated.

This is the charm of mathematics. The interlaced numbers and symbols are like the words of the devil, but they bring the truth of the world.

Time passed little by little, and the manuscript paper on the table was gradually covered with black writing.

With a clear idea, it is not difficult for Xu Chuan to write out the proof process smoothly.

Even if he encounters some mathematical calculations during the writing process, it is only to stop him for a few minutes.

On the other side, the buddy who had just written a title for his graduate thesis stretched himself and was about to go to dinner.

Suddenly, Xu Chuan who was constantly writing on the side caught his attention.

This person was here when he came at six o'clock in the morning, and now at six o'clock in the evening, he was ready to go to dinner, and this person was still sitting here, which aroused his curiosity.

Looking at the thick hair and the somewhat immature face, he should be an undergraduate, right?

But what kind of question is this, functional analysis or real variable function? It's been a day and it's still not done?

Although he was curious, he didn't bother others. He deliberately slowed down a little when passing by, so as not to disturb the junior, and at the same time glanced at the manuscript paper on the table.

If it is about functional analysis or real variable functions, which are undergraduate content, he should be able to help this elementary school student, and by the way, pretend to be a β in front of the newcomer.

Read on, read on, read on.

The important thing is said three times. We have already entered the fourth round of recommendation. In the next round, we can go to PK to apply for Sanjiang. Sanjiang has always been the dream of Yaoo, please everyone.