Math 412-501Theory of Partial Differential EquationsLecture 3-3: Bessel functions.Spectral properties of the Laplacian in a circleEigenvalue problem:∇2φ + λφ = 0 in D = {(x, y) : x2+ y2≤ R2},u|∂D= 0.In polar coordinates (r, θ ):∂2φ∂r2+1r∂φ∂r+1r2∂2φ∂θ2+ λφ = 0(0 < r < R, −π < θ < π),φ(R, θ) = 0 (−π < θ < π).Additional boundary conditions:|φ(0, θ)| < ∞ (−π < θ < π),φ(r, −π) = φ(r, π),∂φ∂θ(r, −π) =∂φ∂θ(r, π) (0 < r < R).Separation of variables: φ(r, θ) = f (r)h(θ).Substitute this into the equation:f′′(r)h(θ) + r−1f′(r)h(θ) + r−2f (r)h′′(θ) + λf (r)h(θ) = 0.Divide by f (r)h(θ) and multiply by r2:r2f′′(r) + r f′(r) + λr2f (r)f (r)+h′′(θ)h(θ )= 0.It follows thatr2f′′(r) + r f′(r) + λr2f (r)f (r)= −h′′(θ)h(θ )= µ = const.The variables have been separated:r2f′′+ rf′+ (λr2− µ)f = 0,h′′= −µh.Boundary conditions φ(R, θ) = 0 and |φ(0, θ)| < ∞hold if f (R) = 0 and |f (0)| < ∞.Boundary conditions φ(r, −π) = φ(r, π) and∂φ∂θ(r, −π) =∂φ∂θ(r, π) hold if h(−π) = h(π) andh′(−π) = h′(π).Eigenvalue problem:h′′= −µh, h(−π) = h(π), h′(−π) = h′(π).Eigenvalues: µm= m2, m = 0, 1, 2, . . . .µ0= 0 is simple, the others are of multiplicity 2.Eigenfunctions: h0= 1, hm(θ) = cos mθ and˜hm(θ) = sin mθ for m ≥ 1.Dependence on r:r2f′′+ rf′+ (λr2− µ)f = 0, f (R) = 0, |f (0)| < ∞.We may assume that µ = m2, m = 0, 1, 2, . . . .Also, we know that λ > 0 (Rayleigh quotient!).New variable z =√λ · r removes dependence on λ:dfdr=√λdfdz,d2fdr2= λd2fdz2.z2d2fdz2+ zdfdz+ (z2− m2)f = 0This is Bessel’s differential equation of order m.Solutions are called Bessel functions of order m.z2d2fdz2+ zdfdz+ (z2− m2)f = 0Solutions are well behaved in the interval (0, ∞).Let f1and f2be linearly independent solutions.Then the general solution is f = c1f1+ c2f2, wherec1, c2are constants.We need to determine the behavior of solutions asz → 0 and as z → ∞.In a neighborhood of 0, Bessel’s equation is a smallperturbation of the equidimensional equationz2d2fdz2+ zdfdz− m2f = 0.Equidimensional equation:z2d2fdz2+ zdfdz− m2f = 0.For m > 0, the general solution isf (z) = c1zm+ c2z−m, where c1, c2are constants.For m = 0, the general solution isf (z) = c1+ c2log z, where c1, c2are constants.We hope that Bessel functions are close to solutionsof the equidimensional equation as z → 0.Theorem For any m > 0 there exist Besselfunctions f1and f2of order m such thatf1(z) ∼ zmand f2(z) ∼ z−mas z → 0.Also, there exist Bessel functions f1and f2of order0 such thatf1(z) ∼ 1 and f2(z) ∼ log z as z → 0.Remarks. (i) f1and f2are linearly independent.(ii) f1is determined uniquely while f2is not.Jm(z): Bessel function of the first kind,Ym(z): Bessel function of the second kind.Jm(z) and Ym(z) are certain linearly independentBessel functions of order m.Jm(z) is regular while Ym(z) has singularity at 0.Jm(z) and Ym(z) are special functions.As z → 0, we have for m > 0Jm(z) ∼12mm!zm, Ym(z) ∼ −2m(m − 1)!πz−m.Also, J0(z) ∼ 1, Y0(z) ∼2πlog z.Jm(z) is uniquely determined by its asymptotics asz → 0. Original definition by Bessel:Jm(z) =1πZπ0cos(z sin τ − mτ) dτ .Behavior of the Bessel functions as z → ∞ doesnot depend on the order m. Any Bessel function fsatisfyf (z) = Az−1/2cos(z −B) + O(z−1) as z → ∞,where A, B are constants.The function f is uniquely determined by A, B, andits order m.As z → ∞, we haveJm(z) =r2πzcosz −π4−mπ2+ O(z−1),Ym(z) =r2πzsinz −π4−mπ2+ O(z−1).Let 0 < jm,1< jm,2< . . . be zeros of Jm(z) and0 < ym,1< ym,2< . . . be zeros of Ym(z).Then the zeros are interlaced:m < ym,1< jm,1< ym,2< jm,2< . . . .Asymptotics of the nth zeros as n → ∞:jm,n∼ (n +12m −14)π, ym,n∼ (n +12m −34)π.Eigenvalues of the Laplacian in a circleIntermediate eigenvalue problem:r2f′′+ rf′+ (λr2− m2)f = 0, f (R) = 0, |f (0)| < ∞.New variable z =√λ · r reduced the equation toBessel’s equation of order m. Hence the generalsolution is f (r) = c1Jm(√λ r) + c2Ym(√λ r),where c1, c2are constants.Singular condition |f (0)| < ∞ holds if c2= 0.Nonzero solution exists if Jm(√λ R) = 0.Thus there are infinitely many eigenvalues λm,1, λm,2, . . . ,wherepλm,nR = jm,n, i.e., λm,n= (jm,n/R)2.SummaryEigenvalue problem:∇2φ + λφ = 0 in D = {(x, y) : x2+ y2≤ R2},u|∂D= 0.Eigenvalues: λm,n= (jm,n/R)2, wherem = 0, 1, 2, . . . , n = 1, 2, . . . , and jm,nis the nthzero of the Bessel function Jm.Eigenfunctions: φ0,n(r, θ) = J0(j0,nr/R).For m ≥ 1, φm,n(r, θ) = Jm(jm,nr/R) cos mθ and˜φm,n(r, θ) = Jm(jm,nr/R) sin
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