Chapter 73 Proof of weakening Weyl_Berry conjecture
After chatting with Zhou Hai about the Weyl-Berry conjecture in the classroom, Xu Chuan locked himself in the library again.
What I have to say is that although the Weyl-Berry conjecture is a world-class conjecture and can even be ranked as difficult as T3, there is really not much information about this conjecture.
However, as he researched, Xu Chuan unexpectedly discovered that the first asymptotic theorem of the Weyl conjecture, the predecessor of the Weyl-Berry conjecture, was 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 studied in the library about the Weyl-Berry conjecture.
Starting from the elliptic operator, to the differential operator and then to the Laplace operator, Xu Chuan did not miss every basic book related to the Weyl-Berry conjecture.
In the library, Xu Chuan closed the books in his hands, took out his laptop from his bag, created a new document, and wrote:
[Proof of spectral asymptotic and weak Weyl_Berry conjecture on connected regions with fractal boundaries! 】
A long period of study, coupled with the mathematical knowledge he brought back from his rebirth, gave him a deep understanding of spectral asymptotics in connected regions with fractal boundaries.
Although it is currently not possible to directly prove the Weyl_Berry conjecture, Xu Chuan thought he could give it a try by weakening the Weyl_Berry conjecture and making it a type of naturally connected fractal drum that satisfies the 'cut' condition.
At least in this area, he already has some ideas in his mind, and he can write them out regardless of whether he can succeed or not.
[Introduction: In 1993, Lapidy and Pomerance 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 This is generally no longer true under the Minkowski framework. 】
[But at the same time, two mathematicians, Levitin M and Vasilyev, proved that under a special type of high-dimensional examples, the Weyl-Berry conjecture is true 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 such that the boundary Ω is measurable under this fractal framework, and the Weyl-Berry conjecture is established under this fractal framework?"]
After writing the title and introduction, Xu Chuan skipped the text and typed out a few lines of blank spaces.
Cited documents:
[[1]KigamiJ, LapidusML.Weyl on the spectral distribution of the Laplacian operator, P.C.F. Self-similar sets. Journal of Mathematics and Physics, 1993, 158:93-125]
[[2] Spectral asymptotic, update theorem and Berry conjecture for a class of fractals. Journal of Mathematics and Engineering, 1996, 72(3):188-214]
【.】
There are not many cited documents, less than a slap in the face.
This can only be said that few people have made much tangible contributions in this area.
In fact, this is exactly the case. Since 1979, when M.V. Berry, a physicist in the land 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, in the past few decades, Countless 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, except for the fact that in 1993, two mathematicians, Lapidy and Pomerance, proved that the one-dimensional Weyl-Berry conjecture is true, there has been almost no new results.
Countless mathematicians, mathematics enthusiasts and physicists have spent more than thirty years of hard work, but no one has been able to successfully turn the Weyl-Berry conjecture into the Weyl-Berry theorem.
But here lies the charm of mathematics and physics. Each conjecture is like a heavy fruit hanging on the tree. Whether you are a mathematician or a physicist, you can see the attractive bright red and plump fruit shape. All that is waiting is for a mathematician or physicist to build a ladder and climb up to pick it up.
Well, the exception was Uncle Newton. Others climbed up with a ladder to pick it, but for him the apple fell down and hit his head.
After typing down the title and introduction, Xu Chuan put the computer aside, took out a stack of A4 manuscript paper from his schoolbag, and began to continue writing down the ideas in his mind.
The library of NTU 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 lazily ran back to the dormitory.
Assuming ΩRn is a bounded open set, we consider the following eigenvalue problem of the Dirichlet-Laplace operator: (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<λ1≤λ2≤λk≤. . . . .
Here limk→+∞λk=+∞, the question we are interested in is which geometric quantities of Ω are spectrally invariant (that is, they are uniquely determined by the spectrum {λi}i∈N).
Problems in this area rely on studying the asymptotic behavior of the eigenvalue λk as k→+∞. For λ>0, it is defined.
The black pen in his hand continued to outline symbols and words on the white manuscript paper.
For Xu Chuan, when he entered the proof process, he had already ignored everything around him. Everything in the world no longer existed in his eyes. There were only the manuscript paper and pen on the table, and the lines of calculations and equations output from his mind. Word.
When numbers, theorems, formulas and symbols dance under the tip of the pen, the beauty brought by the perfect rhythm constantly emerges in Xu Chuan's mind, making him intoxicated.
This is the charm of mathematics. The interlaced numbers and symbols are like the devil's words, 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 in mind, 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 only blocks his time for a few minutes.
On the other side, the buddy who had just written a title for his graduate thesis stretched and prepared to go to dinner.
Suddenly, Xu Chuan, who was writing continuously on the side, caught his attention.
This man was here when he came at six in the morning. Now at six in the evening, he was ready to have dinner. This man was still sitting here, which aroused his curiosity.
Judging from the thick hair and the somewhat immature face, he must be an undergraduate, right?
But what kind of problem is this, functional analysis or real variable function? It's been a day and you still haven't finished it?
Although he was curious, he did not disturb others. When passing by, he deliberately slowed down a little to avoid disturbing the junior student and at the same time glanced at the manuscript paper on the table.
If it were undergraduate content such as functional analysis or real variable functions, he should be able to help this junior student, and by the way, pretend to be beta in front of the newcomers.
Please pursue reading, please pursue reading, please pursue reading.
Important things have been said three times. We have now entered the fourth round of recommendations. In the next round, you can go to PK to apply for Sanjiang. Sanjiang has always been the dream of Yawei, please.
(End of chapter)