Differential Operators and Giant Fiery Laser Beams and Commuting Operators for Department of Mathematics University of Washington Winter 2015
Differential Operators and Outline 1 Differential Operators and
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
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?
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!
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).
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
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
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.
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.
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 = 1 2 + ik, c = 1 + 1 4 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)
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.
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!
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. 39-90. Mulase, Motohico. "Algebraic Theory of the KP Equations". Perspectives in Mathematical Physics (1994), pp 151-218