After talking to 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 even the difficulty can be ranked around T3, there is really not much information about this conjecture.

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

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

Sure enough, math and physics go hand in hand!

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

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

.......

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

[Proof of spectral asymptotic and weak Weyl_Berry conjecture on a connected region with fractal boundaries!] 】

The long period of study, coupled with the mathematical knowledge brought back by rebirth, gave him a deep enough understanding of the spectral asymptotic on the connected region with fractal boundaries.

Although it is not possible to directly prove Weyl_Berry conjecture at present, after weakening the Weyl_Berry conjecture and making it meet the 'cut' condition, the connected fractal drum is a type of naturally connected fractal drum Xu Chuan felt that he could give it a try.

At least in this part, he already has some ideas in his heart, and he can write them out whether he can succeed or not.

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

But at the same time, Levitin The mathematicians M and Vasilyev also proved that the Weyl-Berry conjecture holds within the Minkowski framework for a special class of high-dimensional examples. 】

[All of this shows that the use of the Minkowski framework does not cover all the complexities of the problem, so the correct formulation of the Weyl-Berry conjecture should be:

"Is there a fractal frame that makes the boundary ∂Ω measurable under this fractal frame, and the Weyl-Berry conjecture holds within this fractal frame?" 】

After writing down the title and introduction, Xu Chuan skipped the text and knocked down a few lines of space.

Citations:

[[1]Kigami J, Lapidus M L. Weyl on the problem of the distribution of the Laplace operator spectrum, P. C. F. Self-similar set. Acta Mathematica Sinica, 1993, 158: 93-125.

[[2] Spectral asymptotic, updating theorems and Berry's conjecture for a class of fractals. Chinese Journal of Mathematics and Engineering, 1996, 72(3): 188-214]

【.....】

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

It can only be said that there are hardly many people who have made much contribution in this area.

In fact, it is precisely this since 1979, the physicist M. After V. Berry extended the Weyl conjecture to the case where the Ω is a fractal region while studying the scattering of light waves on fractal objects, countless mathematicians and mathematics enthusiasts, as well as physicists, have worked on spectral asymptotic regions on the connected regions with fractal boundaries.

Thirty years later, apart from 1993, two mathematicians, Lapidi and Pomerance, proved that the one-dimensional Weyl-Berry conjecture was true, there were few new results.

Countless mathematicians, math enthusiasts, and physicists have worked hard for more than 30 years, but none of them have succeeded in turning the Weyl-Berry conjecture into the Weyl-Berry theorem.

But the charm of mathematics and physics lies here, and one by one conjectures hang on the tree like heavy fruits, and both mathematicians and physicists can see the alluring red and full fruit shape.

All that's 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 climb up with a ladder to pick it, and he is the apple himself who falls and hits his head.

......

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

The library of NTU is large, and some areas are quite quiet.

Just like where he is now, because the books stored are all relatively remote books, and there are not many people around, Xu Chuan is too lazy to run back to the dormitory.

...... Let Ω⊂ Rn be a bounded open set, and we consider the eigenvalue problem of the Dirichlet-Laplace operator as follows: (P){-△ u=λu, x∈Ω; u|∂Ω= 0

Then problem (P) has a discrete spectrum {λi}i∈N, and can be arranged in a column: 0 0, defined ......

The black pen in his hand kept outlining symbols and words on the white manuscript paper.

For Xu Chuan, after entering the process of proof, he has ignored everything around him, 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 words that are output from his mind.

When numbers and theorems, formulas and symbols dance under the tip of the pen, the beauty of that perfect beat continues to emerge in Xu Chuan's heart, and he is intoxicated.

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

As time passed, the manuscript paper on the table gradually became covered with black handwriting.

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 in the process of writing, it is only a few minutes to stop him.

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

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

This man was here when he came at six o'clock in the morning, and now it was six o'clock in the evening, and he was ready to go to dinner, and this man was still sitting here, which could not help arousing his curiosity.

Looking at this thick hair and still a little immature face, it should be an undergraduate, right?

But what kind of question is this, functional analysis or real function? It's been a day and you haven't done it yet?

Although he was curious, he didn't bother others, and 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 functional analysis or real function for undergraduates, he should be able to help this junior brother, and by the way, put on a β in front of the newcomer.

.....