Some Mathematical and Physical Background
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1 Some Mathematical and Physical Background Linear partial differential operators Let H be a second-order, elliptic, self-adjoint PDO, on scalar functions, in a d-dimensional region Prototypical categories Billiard: H =, R d, boundary conditions (say Dirichlet, u =0on ) Example: = rectangular parallelepiped in R 3 ( brick ), H = x + y + z, u(0,y,z)=u(l 1,y,z)=u(x, 0,z)=u(x, L,z) =u(x, y, 0) = u(x, y, L 3 )=0 1
2 Laplace Beltrami operator: =d-dimensional Riemannian manifold (without boundary), Hu = g u = d j,k=1 1 g x j [ gg jk u ] x k Example: = -sphere, Hu = 1 sin θ θ ( sin θ u ) θ 1 sin θ u φ Schrödinger operator: =R d (say), H = + V (x) Example: Harmonic oscillator (d = 1), H= d dx + x Ignoring: external magnetic potential Neumann and Robin boundary conditions fourth-order operators, etc pseudodifferential operators vector field u (electromagnetism) first-order elliptic system (Dirac field)
3 Spectral decomposition All the examples so far have discrete spectrum Eigenvalues and normalized eigenvectors: Hϕ n = E n ϕ n, ϕ n = ϕ n (x) dx =1 u(x)= c n ϕ n (x), c n = ϕ n,u = f(h)u = f(e n )c n ϕ n In other words, H is diagonal in this basis ϕ n (x)u(x)dx Brick: ϕ mnp = Examples 3 sin mπx L 1 L L 3 L 1 sin nπy L sin pπz L 3, E mnp = ( mπ L 1 ) ( ) ( nπ pπ + + L L 3 ) (m, n, p Z + ) Sphere: E l = l(l +1),l N,m= l,, l 1, l ϕ lm = Y m l (θ, φ) = (spherical harmonic) [ l+1 4π (l m)! (l + m)! ] 1 P m l (cos θ)e imφ 3
4 Harmonic oscillator: E n =n+1 (n N), ϕ n (x) =( π n n!) 1 e 1 x H n (x) (Hermite function) Stadium (classically chaotic billiard): 0 <E 0 <E 1 E E 3 E n + Eigenfunctions ϕ n have been computed numerically up to n >
5 Integral kernels (Green functions) and their traces f(h)u = f(e n ) ϕ n,u ϕ n At least formally, f(h)u(x) = G(x, x)u( x)d x, G(x, y) = f(e n )ϕ n (x)ϕ n (y) If f is sufficiently rapidly decreasing, this converges to a smooth function Trace: G(x, x) dx = f(e n ) Classic example: Heat kernel, f t (H) =e th u(t, x) = K(t, x, y)u 0 (y) dy u t = H (x)u, solves u(0,x)=u 0 (x) K(t, x, y) = e te n ϕ n (x)ϕ n (y) 5
6 Theorem 1: For compact manifolds and compact billiards, with H = g + V (x) (and assuming all E n 0), there is an asymptotic expansion as t 0, K(t, x, x) dx (4πt) d/ a m/ t m/, m=0 where a s is an integral of geometrical data over or its boundary (When s is half-integral (m odd), a s comes entirely from the boundary) In particular, for Dirichlet boundary conditions, a 0 = = d-dimensional volume of, π = (surface area, ), ( a 1 = 1 6 R V ) dx κdσ a 1 (R = Ricci scalar curvature, κ = extrinsic curvature (trace of second fundamental form)) This theorem assumes all the geometrical functions are smooth For the brick we will find directly instead the same a 0 and a 1 as above, but a 1 = π(l 1 + L + L 3 ), a3 = π3/ 6
7 Implications: 1 Inverse problem: From K (hence from {E n }) we can deduce aspects of the geometry of Tauberian problem: From K (hence from and V ) we can deduce aspects of the spectrum {E n } by taking the inverse Laplace transform Theorem (Weyl s Law): Let N(E) bethenumber of eigenvalues E Under conditions of Theorem 1, as E N(E) (4π) d/ Γ(1 + d/) Ed/ Formal check: e te n = 0 e te dn(e) (4π) d/ d e te E d 1 de 0 Γ(1 + d/) ( ) d (4π) d/ d =Γ Γ(1 + d/) =(4π) d/ K(t, x, x) dx 7
8 Examples N(E) (4π) d/ Γ(1 + d/) Ed/ Brick: Theorem predicts N(E) L 1L L 3 6π E 3/ n In fact, by counting lattice points (m, n, p) inoctant of an ellipsoid (handling the coordinate planes by inclusion-exclusion principle) one approximates N(E) V 6π E3/ where V =volume=l 1 L L 3, S 16π E + p L 16π E1/ C 64, S= surface area = (L 1 L + L 1 L 3 + L L 3 ), L = total edge length = 4(L 1 + L + L 3 ), C = number of corners = 8 Laplace transform agrees exactly with (promised) heat kernel expansion! 8
9 Sphere: Theorem predicts N(E) E Well, N(E) = L (l +1) L, where E = L(L+1) L l=0 Harmonic oscillator: Theorem predicts N(E) E 1, but the eigenvalues are equally spaced, so actually N(E) E The region is not compact, so Theorem 1 and the simplest version of Weyl s formula do not apply! The inverse Laplace transform of the heat-kernel expansion cannot be evaluated term-by-term beyond the leading (Weyl) term N(E) is a step function (and N (E) a series of delta functions), so the expansion N(E) E d/ m=0 (4π) d/ Γ(1 + (d m)/) E m/ a m/ must be literally false beyond order m = d at best However, it is correct in some averaged sense (I tacitly averaged N(E) for the brick by not counting lattice points carefully near the ellipsoidal surface) 9
10 Some popular averaging methods 1 Riesz means: Integrate N(E) p times, then divide by the volume of the p-simplex you integrated over p =:! E E 0 de 1 E1 0 de N(E ) (Related to Cesàro summation of series) Then the first p + 1 terms of the formal series in 1/ E (for this new object) are rigorously asymptotic Lorentzian smoothing: (ρ = N ) ρ(e) = ɛ π ρ(e 1 ) de 1 (E E 1 ) + ɛ (Equivalent to analytically continuing the resolvent kernel into the complex plane) For sufficiently large ɛ, ρ has a valid asymptotic expansion in 1/ E 3 Gaussian smoothing (better in high dimensions) 10
11 Continuous spectrum Example: Half-line 0 <x< with Dirichlet BC Here we have a generalized eigenfunction expansion, the Fourier sine transform: u(x) = 0 c k ϕ k (x)dk, c k = ϕ k (x) = π sin(kx) / L (0, ) 0 ϕ k (x)u(x) dx, For general operators it may be more convenient to start with the heat kernel (which exists by abstract PDE or operator theory) and consider its (exact) inverse Laplace transform, which will be the integral kernel of the spectral projection operator That is, if K(t, x, y) = 0 e te dp (E,x,y), then P (E,x,y) is the integral kernel of P (E), the orthogonal projection onto the part of L corresponding to spectrum E (If you don t know Stieltjes integration, think of dp (E) asp (E)dE) When the spectrum is discrete, we just have P (E,x,y)= ϕ n (x)ϕ n (y) n:e n E 11
12 In the half-line example, equivalent to P (E,x,y)= E 0 π sin(kx)sin(ky) dk, K(t, x, y) =(4πt) 1/ [ e (x y) /4t e (x+y) /4t ] P(E,x,x) isalocal spectral density Analogs of Theorems 1 and hold for it and K(t, x, x), but the expansions are nonuniform in x and can t be integrated up to the boundary to recover the integrated quantities a s etc For a billiard, at any interior point K(t, x, x) (4πt) d/ + O(t ), but we know the integral Kdxcontains a boundary term O(t (d 1)/ ) For the half-line, L L K(t, x, x) dx =(4πt) 1/ 0 j=1 0 [ ] 1 e x /t dx ) πt ( (4πt) 1/ L 1 3 ( So for the brick, we get (4πt) 1/ L j ) πt = (4πt) 3/ [ V S πt + Lπt 4 C(πt)3/ 8 ], as claimed 1
13 The main game the operator: potentials geometry classical mechanics integral kernels spectral apparatus: eigenvalues eigenfunctions spectral projections local spectral densities vacuum energy Old spectral asymptotics: (this talk) gross geometry heat trace at small t average eigenvalue density at high frequency New spectral asymptotics: (rest of workshop) classical mechanics kernels details of spectrum 13
14 Heat: f t (H) =e th ; A variety of kernels K t = HK, Quantum: f t (H) =e ith ; (analytic continuation of K) Resolvent: f z (H) = 1 N (E)=ρ(E)= 1 π H z ; K(0,x,y)=δ(x y) i U t = HU, U(0) = δ (H z)g = δ(x y) Im G(E + iɛ,x,x)dx Zeta: f s (H) =H s (meromorphic function of s) Wave: D Alembert: f t (H) =cos(t H); D t = HD, D(0,x,y)=δ(x y), D t (0) = 0 H Wightman: f t (H) = e it H (positive frequency solution) Cylinder: f t (H) =e t H ; T =+HT, T(0,x,y)=δ, T 0ast t (related to wave kernels by analytic continuation) 14
15 Physical interpretations and physical quantities (What happened to h, m, c, and all that?) Schrödinger equation: i h u t = h m u + λv (x)u 1 u ( mc ) Klein Gordon equation: c t = u u h (Nonrelativistic and relativistic meanings of m are different!) Wave equation is case m = 0; instead, could ( mc ) generalize to V (x)u h Can choose time unit and redefine constants so i u t = u + λv u Hu, u t = u + λv u Separation of variables (respectively, u = ϕe iet/ h,or u=ϕe iωt/ h with E = ω ), leads always to Hϕ n = E n ϕ n There are two independent constants, λ and E Butit is customary and convenient to reintroduce h, as we ll see 15
16 The hierarchy of physical theories classical mechanics geometrical optics (ODE systems) semiclassical approximation nonrelativistic quantum mechanics (Schrödinger equation) classical field theory (wave equation) large mass many particles relativistic quantum field theory (Fock space, etc) superstrings or something 16
17 Semiclassical and high-frequency limits Let s set Schrödinger m = 1 and Klein Gordon c =1, but leave h visible In either case, h ϕ + λv ϕ = Eϕ There are two similar limits: High frequency: E +, hand λ fixed Semiclassical: h 0 equivalent to E and λ simultaneously In a billiard (V = 0) these are the same! Underlying classical system: p i,so H=p +λv (x) dx dt = H p =p, dp dt = H x Thus, Newton s law of motion: = λ V (x) d x dt = λ V (x) = F m (recall m = 1 ) Semiclassical approximation is built on the classical trajectories High-frequency approximation is built on force-free classical trajectories (straight rays) 17
18 Field quantization and vacuum energy Consider the wave equation in a billiard, u t = u, u =0on Expand in the normal modes: u(t, x) = c n (t)ϕ n (x), Hϕ n = ω n ϕ n Now treat each mode amplitude c n as a quantum system It behaves as a harmonic oscillator: state N n with N n quanta (N n =0,1,) whose energy is (N n + 1 ) hω n Vacuum state is N n =0 n Set h = 1 Total energy E = 1 ω n =? m=1 Z(s) = (ω n ) s, E = 1 Z( 1 ), defined by analytic continuation 18
19 One-dimensional billiard yields Riemann zeta function: ω n = nπ L, Z ( 1 ) = π L n = π 1L =E L dependence de dl = π > 0, attractive force 4L 19
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