Giant Fiery Laser Beams and Commuting Operators

Transcription

1 Differential Operators and Giant Fiery Laser Beams and Commuting Operators for Department of Mathematics University of Washington Winter 2015

2 Differential Operators and Outline 1 Differential Operators and

3 Basic Question Differential Operators and Question What differential operators commute with a the following operator δ = 2 + rsech 2 (x) Some basic observations: Commuting operators form an algebra C(δ) Any polynomial in δ commutes with δ Operators commuting with δ must have constant leading coefficient Algebra C(δ) = C[δ] or C(δ) generated by δ and an operator of odd order

4 Example 1 Differential Operators and Example If r = 0, δ = 2 and C(δ) = C[ ] (affine line). Example If r = 2, δ = 2 and C(δ) = C[δ, η] (singular cubic), for η the order 3 operator given by η = 3 (3sech 2 (x) 1) + 3sech 2 (x) tanh(x). Can we say something about the algebra in general?

5 Differential Operators and Helpful Theorems Theorem (Schur) For any differential operator ω, the algebra C(ω) is commutative. Theorem (Burchnall-Chaundy) For any second-order differential operator ω, the algebra C(ω) is an irreducible algebraic plane curve. Theorem For δ as given by the question, the curve C(δ) is rational. These theorems give us a general framework, but we can say more!

6 Differential Operators and Schrödinger Operators Definition A differential operator of the form 2 + q(x) is called a Schrödinger operator. The function q(x) is called the potential. A potential is called short-range if q(x) goes to zero rapidly as x ±. These definitions come from (steady-state) Schrödinger s equation 2 2m ψ (x) + v(x)ψ(x) = Eψ(x). describing a one-dimensional particle interacting with a potential v(x).

7 Differential Operators and Figure: A mighty laser blast revealing molecular structure (Credit: SLAC National Accelerator Laboratory) to understand shape of things too small to see, shoot small particles at it and see where they end up figuring out where a particular potential will send particles is direct scattering recovering the structure of a potential from the scattered particles is inverse scattering

8 Differential Operators and Beam Fired from Far Away A laser fired from sends a particle at a one-dimensional short-range potential Near +, we see transmitted beam Near, we see fired beam + reflected beam (opposite phase) Away from potential, particles in beam are free particles" Figure: A sketch of wave function from a beam at

9 Jost Solutions Differential Operators and The wave function of a particle in a beam from has the asymptotic behavior { T (k)e ψ(x) = ikx, x e ikx + R(k)e ikx, x ±k depends on beam frequency and phase The right irregular Jost solution f ± (x, k, ) is unique eigenfunction of δ with eigenvalue k 2 satisfying f ± (x, k, ) = e ±ikx + o(1/x) as x. The left irregular Jost solution f ± (x, k, ) is defined similarly, but satisfies f ± (x, k, ) = e ±ikx + o(1/x) as x.

10 Data Differential Operators and Note ψ(x) = T (k)f + (x, k, ) (for 2 /2m = 1) f + (x, k, ) = T (k) 1 f + (x, k, ) + T (k) 1 R(k)f (x, k, ) T (k) and R(k) are transmission and reflection coefficients We call a short-range potential v(x) reflectionless if R(k) = 0 for all k Theorem If a Schrödinger operator with short-range potential commutes with an operator of odd order, then the potential must be reflectionless.

11 Differential Operators and for rsech 2 (x) after a change of variable and gauge transformation, δ becomes the hypergeometric operator 2 z (1 z)z + z (c (a + b + 1)z) ab a = 1 2 ik, b = ik, c = r. can calculate eigenfunctions in terms of hypergeometric functions gives us scattering coefficients T (k) = Γ(c b)γ(2a + b c) Γ(1 ik)γ( ik) T 1 (k)r(k) = sec(πk) 1 sec(π 4 r)

12 Centrilizer of δ Differential Operators and Corollary The potential v(x) = rsech 2 (x) is reflectionless if and only if r = n(n + 1) for some nonnegative integer n. Theorem If r = n(n 1), then C(δ) is generated by an operator of order 2 and an operator of order 2n + 1, and is isomorphic to C[u, v]/(v 2 f (u) 2 u), f (u) = n (u m 2 ), m=1 which is a rational plane curve of degree n with n double points.

13 Summary! Differential Operators and Calculating centrilizers of differential operators is complicated and fun! Differential operators give rise to interesting algebraic constructions Giant Fiery Laser Beams is a Great Title Thanks for coming!

14 References: Differential Operators and Koelink, Erik. " Theory" Landelijke Master Radboud Universiteit Nijmegen, Spring 2008 Segal, Graeme and George Wilson. "Loop Groups and Equations of KdV Type". Comm. on Pure and Applied Mathematics 37 (1984), pp Mulase, Motohico. "Algebraic Theory of the KP Equations". Perspectives in Mathematical Physics (1994), pp