Mathematicians from Peking University and Nankai University Solve the Famous "Ten-Martini Problem": A More Unified and Elegant Proof

A problem that has puzzled the intersection of mathematics and quantum mechanics for half a century finally has a relatively perfect answer thanks to the participation of mathematicians from Peking University and Nankai University.

This problem has a very interesting name, called "The Ten Martini Problem".

The reason for this name is that in 1981, mathematician Mark Kac said that he would buy ten martinis for whoever could solve this problem.

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To put it simply, the Ten Martini Problem is a conjecture about the energy spectrum structure of quantum systems. It asserts that the energy spectrum of "Almost Mathieu operators" at all irrational frequencies is a Cantor set.

Among them, "Almost Mathieu operators" are special Schrödinger operators with a cosine potential; a Cantor set is a fractal structure (it looks like "dust", with no intervals and only infinitely dispersed points).

Although between 2004 and 2005, mathematicians Avila and Jitomirskaya finally gave a complete proof that the energy spectrum of "Almost Mathieu operators" is indeed a Cantor set (Avila later won the Fields Medal for this).

But when two Chinese mathematicians (Ge Lingrui from Peking University and You Jiangong from Nankai University) joined Jitomirskaya's research, they further generalized this conclusion:

Not only for "Almost Mathieu operators", but also for a larger class of "quasiperiodic operators", a similar Ten Martini property holds, that is, the energy spectrum is a Cantor set.

Their work not only provides an unprecedentedly elegant and unified proof for this classic problem, but more importantly, it generalizes the conclusion from a highly idealized model to a broader scenario closer to real physical systems.

But to fully understand the story of the Ten Martini Problem, we need to go back to 1974.

A Physics Conjecture Born from "The Butterfly on the Calculator"

The story starts with a man named Douglas Hofstadter.

At that time, he was just a physics graduate student at the University of Oregon in the United States. That year, he went to Regensburg, Germany, with his supervisor for an academic sabbatical and joined a research group of top theoretical physicists.

(PS: Five years later, he would win the Pulitzer Prize for writing "Gödel, Escher, Bach: An Eternal Golden Braid" and become a world - renowned thinker.)

The core topic of the group was a fundamental and tricky quantum problem -

When an electron moves in a regularly arranged crystal (lattice) and is simultaneously affected by a perpendicular magnetic field, how will its energy be distributed?

This problem is known in physics as "the energy spectrum problem of Bloch electrons in a magnetic field". The physicists in the group generally used rigorous and abstract theorem deductions, trying to directly derive the final conclusion theoretically.

But Hofstadter found it difficult to follow his colleagues' advanced mathematical thinking. He recalled:

To some extent, my luck was that I couldn't follow them. They were proving theorems, but those theorems seemed irrelevant to the essence of the problem.

So, Hofstadter decided to take a different approach and adopt an experimental method that seemed rather "clumsy" at that time: numerical calculation.

He found a Hewlett - Packard 9820A desktop calculator, a device weighing nearly 40 pounds, with functions between a calculator and an early computer.

His plan was not to directly tackle the most difficult theoretical situation, but to start with a simplified problem and "see" the answer through a large number of calculations.

The core tool for describing this physical system is the cornerstone of quantum mechanics - the Schrödinger equation.

In this specific problem, the "energy spectrum" obtained from the equation, that is, the set of energy values that an electron is allowed to have, is determined by a key parameter α. This parameter α is proportional to the product of the magnetic field strength and the unit area of the lattice, and it captures the essence of the influence of the external magnetic field on the electron's motion.

Theoretically, when α is a rational number, the system is periodic. Although the calculation is cumbersome, it is solvable in principle. However, when α is an irrational number, the system no longer has simple periodicity and becomes a so - called "quasi - periodic" system, and solving it was a huge theoretical obstacle at that time.

Hofstadter didn't plunge into the theoretical dilemma of irrational numbers like his colleagues. He chose to start from the known rational number situation. He programmed the calculator to automatically calculate the energy spectra corresponding to a series of rational α values.

Every night, he set an α value and let the calculator work overnight. The next morning, he would see a long paper tape coming out of the machine, with the positions of all allowed energy values at that α value printed on it.

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He painstakingly plotted these data points on a large piece of graph paper. The horizontal axis of this graph represents the electron's energy, and the vertical axis represents the parameter α. Each α value corresponds to a horizontal line on the graph, and the marked points on the line are the energy levels at which the electron can exist under that magnetic field strength.

As he plotted more and more dense rational α values, a breathtaking figure gradually emerged.

Between the allowed energy levels (black dots), there are large "forbidden bands" (blank areas), and the shape of these blank areas is amazingly similar to a butterfly with outstretched wings.

More amazingly, this "butterfly" shows clear fractal characteristics: if you zoom in on any small part of the butterfly's wings, you will find that its pattern structure is extremely similar to the shape of the whole butterfly, and this self - similarity can extend infinitely.

This graph was later affectionately called "The Hofstadter Butterfly" by the scientific community.

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His colleagues initially dismissed this "grunt work", and even his supervisor criticized it as "numerical superstition" and threatened to cancel his funding. But Hofstadter firmly believed that there were profound physical and mathematical truths hidden behind this beautiful figure.

By observing the figure, he put forward an amazing conjecture.

He noticed that when the denominator q of the rational number α = p/q as the input becomes larger and larger, that is, the fraction becomes more and more complex and closer to an irrational number, the corresponding energy spectrum band will split into more and more sub - bands, and there will be more and more gaps in between.

Its overall structure visually approaches a famous mathematical object infinitely - that is, the Cantor set we mentioned at the beginning.

A Cantor set is a classic mathematical fractal that can be constructed through a simple iterative process: start with a line segment, remove the middle third of it, leaving two shorter line segments; then, remove the middle third of each of these two line segments respectively; repeat this process infinitely.

What remains in the end will no longer be any line segments, but a set of infinitely many discrete points distributed like dust on the original line segment.

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Hofstadter thus inferred that when the parameter α is a real irrational number, the electron's energy spectrum will no longer be a continuous energy band, but a perfect Cantor set with an infinitely fine structure.

Offering a Reward of "Ten Martinis"

Hofstadter's conjecture was like a stone thrown into a calm lake, causing ripples in the mathematical and physical community.

A few years later, two outstanding mathematicians, Mark Kac and Barry Simon, independently reached the same conclusion as Hofstadter from a pure mathematical perspective when studying a class of mathematical objects called "almost periodic functions".

This greatly increased the credibility of the conjecture, but a strict mathematical proof was still out of reach. The difficulty of the problem inspired Kac's sense of humor.

At the annual meeting of the American Mathematical Society in 1981, he publicly announced, half - jokingly, that he would offer ten martinis to anyone who could strictly prove this conjecture.

Simon then promoted this reward in various academic occasions, making the "Ten Martini Problem" well - known and becoming a yardstick for measuring the progress of quasi - periodic system research.

In the following two decades, generation after generation of mathematicians challenged this difficult problem.

They continuously developed new mathematical tools and successfully proved that the conjecture holds for "certain" specific types of irrational numbers (for example, "Liouville numbers" that can be well - approximated by rational numbers or "Diophantine numbers" with opposite properties).

However, a general proof that can cover all irrational numbers without any loopholes has never appeared.

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Finally, in 2005, the deadlock was broken.

Mathematician Svetlana Jitomirskaya collaborated with the then 24 - year - old genius mathematician Artur Avila and published a landmark paper, announcing that the "Ten Martini Problem" had been solved.

They also happily fulfilled the promise of ten martinis and raised their glasses to celebrate this great victory.

However, after the champagne bubbles of celebration had faded, the "imperfection" of this proof became apparent. It was more like a "patchwork quilt" than a seamless work of art.

Because their proof, in order to cover all irrational numbers, relied on a variety of different and sometimes contradictory technical means: for "mild" Diophantine numbers, they used one set of analysis methods; for "wild" Liouville numbers, they had to resort to another completely different set of algebraic tools.

This made the whole proof process seem complicated and lack internal unified beauty.

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More crucially, this proof has a fundamental limitation: it heavily depends on the special symmetry of the model initially studied by Hofstadter, that is, the so - called "Almost Mathieu Operator".

This model is a highly idealized mathematical object, like a perfect circle in geometry.

But in the real world, physical systems are much more complex: there may be defects in the lattice, and the magnetic field is not absolutely uniform. When people try to generalize this proof to a more realistic model with broken symmetry, the whole proof framework collapses.

This left mathematicians in new confusion. Are the "Hofstadter Butterfly" and the Cantor set just coincidences in that perfect mathematical model?

But a dramatic scene occurred, and the progress of physics once again promoted mathematical thinking.

In 2013, physicists at Columbia， University conducted experiments on two - layer graphene materials under a strong magnetic field and actually observed the energy spectrum structure of the "Hofstadter Butterfly" clearly in the laboratory!

This discovery shocked the academic community and eloquently proved that this strange fractal structure is not a mathematician's castle in the air, but a common and robust physical phenomenon.

Jitomirskaya admitted: "This suddenly turned the problem from a mathematician's imagination into a real thing, which became very disturbing."

Physical reality urgently needs a more powerful and universal mathematical theory for explanation.

Chinese Mathematicians Broke the Deadlock

In 2019, Chinese mathematician Ge Lingrui joined Jitomirskaya's team and worked with You Jiangong, a mathematics professor at Nankai University, to improve Avila's theory.

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This time, they didn't just focus on "Almost Mathieu operators", but defined a broader class of Type I operators, which are defined by T - acceleration.

First, mathematicians added an adjustment parameter to the Lyapunov exponent (a physical quantity describing the "growth/decay rate of solutions" corresponding to quasi - periodic operators). This adjustment parameter will cause the value of the Lyapunov exponent to change, and there will be a turning point between each change.

The T - acceleration can be simply understood as the change rate of the Lyapunov exponent near the first turning point. The specific definition is as follows: