In mathematical physics, the almost Mathieu operator, named for its similarity to the Mathieu operator introduced by Émile Léonard Mathieu, arises in the study of the quantum Hall effect.
Operator Almost Mathieu merupakan operator dalam fisika matematika yang berkaitan dengan studi efek Hall kuantum.
It is given by [ H ω λ , α u ] ( n )
[ H ω λ , α u ] ( n )
u(n),\,} acting as a self-adjoint operator on the Hilbert space ℓ 2 ( Z ) (\mathbb {Z} )} .
u(n),\,} bertindak sebagai operator swa-adjung (self-adjoint operator) pada ruang Hilbert ℓ2(Z).
Here α , ω ∈ T , λ > 0 ,\lambda >0} are parameters.
Parameter yang digunakan terdiri atas α,ω∈T serta λ>0.
In pure mathematics, its importance comes from the fact of being one of the best-understood examples of an ergodic Schrödinger operator.
Dalam matematika murni, operator ini menjadi salah satu contoh operator Schrödinger ergodik yang telah dianalisis secara luas.
For example, three problems (now all solved) of Barry Simon's fifteen problems about Schrödinger operators "for the twenty-first century" featured the almost Mathieu operator.
Tiga masalah dalam daftar lima belas masalah Schrödinger menurut Barry Simon berkaitan langsung dengan operator ini.
In physics, the almost Mathieu operators can be used to study metal to insulator transitions like in the Aubry–André model.
Dalam fisika, operator Almost Mathieu digunakan dalam pemodelan transisi logam–isolator, termasuk dalam Model Aubry-André.
For λ1 , the almost Mathieu operator is sometimes called Harper's equation.
Untuk kasus λ=1, operator ini sering disebut sebagai persamaan Harper.
This problem was named the 'Dry Ten Martini Problem' by Barry Simon as it was 'stronger' than the weaker problem which became known as the 'Ten Martini Problem':
Dugaan ini kemudian dikenal sebagai Dry Ten Martini Problem oleh Barry Simon karena sifatnya lebih kuat daripada versi lemahnya.
For all λ ≠ 0 , all irrational a , and all ω , the spectrum of the almost Mathieu operator is a Cantor set.
Versi lemah dugaan tersebut, yaitu Ten Martini Problem, menyatakan bahwa Untuk setiap λ ≠ 0 , setiap bilangan irasional a , dan setiap ω , spektrum operator Almost Mathieu berbentuk himpunan Cantor.
If α is a rational number, then H ω λ , α ^{\lambda ,\alpha }} is a periodic operator and by Floquet theory its spectrum is purely absolutely continuous.
Jika α merupakan bilangan rasional, operator H ω λ , α ^{\lambda ,\alpha }} bersifat periodik.
Since the transformation ω ↦ ω + α is minimal, it follows that the spectrum of H ω λ , α ^{\lambda ,\alpha }} does not depend on ω .
Jika α merupakan bilangan irasional, transformasi ω ↦ ω + α bersifat minimal sehingga spektrum operator tidak bergantung pada ω .
On the other hand, by ergodicity, the supports of absolutely continuous, singular continuous, and pure point parts of the spectrum are almost surely independent of ω .
Berdasarkan sifat ergodik, dukungan bagi bagian spektrum absolut kontinu, kontinyu singular, dan titik murni hampir pasti tidak bergantung pada ω .
It is now known, that For 0 < λ < 1 , H ω λ , α ^{\lambda ,\alpha }} has surely purely absolutely continuous spectrum.
λ < 1 , H ω λ , α ^{\lambda ,\alpha }} spektrum hampir pasti absolut kontinu.
(This was one of Simon's problems.)
(merupakan salah satu masalah Simon)
For λ1 , H ω λ , α ^{\lambda ,\alpha }} has surely purely singular continuous spectrum for any irrational α .
Untuk λ1 , H ω λ , α ^{\lambda ,\alpha }} spektrum hampir pasti kontinyu singular bagi setiap α irasional.
For λ > 1 , H ω λ , α ^{\lambda ,\alpha }} has almost surely pure point spectrum and exhibits Anderson localization.
Untuk λ > 1 , H ω λ , α ^{\lambda ,\alpha }} spektrum hampir pasti titik murni dan menampilkan lokalisasi Anderson.
(It is known that almost surely can not be replaced by surely.)
(Di ketahui bahwa hampir pasti tidak dapat digantikan dengan pasti.)
That the spectral measures are singular when λ ≥ 1 follows (through the work of Yoram Last and Simon) from the lower bound on the Lyapunov exponent γ ( E ) given by γ ( E ) ≥ max { 0 , log ⁡ ( λ ) } .
Keadaan spektrum yang bersifat singular untuk λ ≥ 1 berikut (melalui karya Yoram Last dan Simon) mengikuti batas bawah eksponen
Another striking characteristic of the almost Mathieu operator is that its spectrum is a Cantor set for all irrational α and λ > 0 .
Operator Almost Mathieu memiliki spektrum berbentuk himpunan Cantor untuk setiap
Furthermore, the Lebesgue measure of the spectrum of the almost Mathieu operator is known to be Leb ⁡ ( σ ( H ω λ , α ) )
Ukuran Lebesgue dari spektrum operator almost Mathieu diketahui memenuhi:
| 4 − 4 λ | (\sigma (H_{\omega }^{\lambda ,\alpha }))
Leb ⁡ ( σ ( H ω λ , α ) )| 4 − 4 λ | (\sigma (H_{\omega }^{\lambda ,\alpha }))
=|4-4\lambda |\,} for all λ > 0 .
=|4-4\lambda |\,} untuk setiap λ > 0 .
For λ1 this means that the spectrum has zero measure (this was first proposed by Douglas Hofstadter and later became one of Simon's problems).
Untuk λ1 hasil ini menyatakan bahwa spektrum memiliki ukuran nol, suatu usulan yang pertama kali diajukan oleh Douglas Hofstadter.
For λ ≠ 1 , the formula was discovered numerically by Aubry and André and proved by Jitomirskaya and Krasovsky.
Untuk λ ≠ 1 , rumus tersebut ditemukan secara numerik oleh Aubry dan André dan dibuktikan secara formal oleh Jitomirskaya dan Krasovsky serta sebagian oleh
The study of the spectrum for λ1 leads to the Hofstadter's butterfly, where the spectrum is shown as a set.
Kajian spektrum untuk λ1 menghasilkan visualisasi yang dikenal sebagai kupu-kupu Hofstadter atau Hofstadter’s butterfly, yang merepresentasikan struktur spektrum sebagai suatu himpunan fraktal.