Asymptotic Analysis and Singular Perturbation Theory

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Energy eigenstates are wavefunctions of the form ψ(t) = e −iEt/ ϕ, where ϕ ∈ H and E ∈ R. It follows from the Schrödinger equation that Hϕ = Eϕ. Hence E is an eigenvalue of H and ϕ is an eigenvector. One of Schrödinger’s motivations for introducing his equation was that eigenvalue problems led to the experimentally observed discrete energy levels of atoms. Now suppose that the Hamiltonian H ε = H0 + εH1 + O(ε 2 ) depends smoothly on a parameter ε. Then, rewriting the previous result, we find that the corresponding simple energy eigenvalues (assuming they exist) have the expansion E ε = E0 + ε 〈ϕ0, H1ϕ0〉 〈ϕ0, ϕ0〉 + O(ε2 ) where ϕ0 is an eigenvector of H0. For example, the Schrödinger equation that describes a particle of mass m moving in R d under the influence of a conservative force field with potential V : R d → R is iψt = − 2 ∆ψ + V ψ. 2m Here, the wavefunction ψ(x, t) is a function of a space variable x ∈ R d and time t ∈ R. At fixed time t, we have ψ(·, t) ∈ L 2 (R d ), where L 2 (R d ) = u : R d → C | u is measurable and R d |u| 2 dx < ∞ is the Hilbert space of square-integrable functions with inner-product 〈u, v〉 = u(x)v(x) dx. R d The Hamiltonian operator H : D(H) ⊂ H → H, with domain D(H), is given by H = − 2 ∆ + V. 2m If u, v are smooth functions that decay sufficiently rapidly at infinity, then Green’s theorem implies that 〈u, Hv〉 = Rd u − 2 ∆v + V v dx 2m 2 2 = ∇ · (v∇u − u∇v) − (∆u)v + V uv dx 2m 2m R d 10
= Rd − 2 ∆u + V u v dx 2m = 〈Hu, v〉. Thus, this operator is formally self-adjoint. Under suitable conditions on the potential V , the operator can be shown to be self-adjoint with respect to an appropriately chosen domain. Now suppose that the potential V ε depends on a parameter ε, and has the expansion V ε (x) = V0(x) + εV1(x) + O(ε 2 ). The perturbation in a simple energy eigenvalue assuming one exists, is given by E ε = E0 + εE1 + O(ε 2 ), E1 = Rd V1(x)|ϕ0(x)| 2 dx Rd |ϕ0(x)| 2 , dx where ϕ0 ∈ L 2 (R d ) is an unperturbed energy eigenfunction that satisfies − 2 2m ∆ϕ0 + V0ϕ0 = E0ϕ0. Example 1.7 The one-dimensional simple harmonic oscillator has potential The eigenvalue problem V0(x) = 1 2 kx2 . − 2 2m ϕ′′ + 1 2 kx2 ϕ = Eϕ, ϕ ∈ L 2 (R) is exactly solvable. The energy eigenvalues are En = ω n + 1 n = 0, 1, 2, . . . , 2 where ω = k m is the frequency of the corresponding classical oscillator. The eigenfunctions are ϕn(x) = Hn(αx)e −α2 x 2 /2 , where Hn is the nth Hermite polynomial, Hn(ξ) = (−1) n dn ξ2 e e−ξ2, dξn 11
Page 1: Asymptotic Analysis and Singular Pe
Page 4 and 5: 4.1.1 Outer solution . . . . . . .
Page 6 and 7: Once we have constructed such an as
Page 8 and 9: For example, setting ε = 0 in (1.2
Page 10 and 11: Solving the first equation, we get
Page 12 and 13: and A ε : R n → R n is a linear
Page 16 and 17: and the constant α, with dimension
Page 18 and 19: The velocity gradient ∇u has dime
Page 20 and 21: Imposing the requirement that R d
Page 23 and 24: Chapter 2 Asymptotic Expansions In
Page 25 and 26: If, as is usually the case, the gau
Page 27 and 28: where denotes the C n -norm. We def
Page 29 and 30: It follows that where Since we have
Page 31 and 32: 2.2.4 Nonuniform asymptotic expansi
Page 33 and 34: Chapter 3 Asymptotic Expansion of I
Page 35 and 36: we find that the optimal truncation
Page 37 and 38: and making the change of variable (
Page 39 and 40: 3.3 The method of stationary phase
Page 41 and 42: Applying this argument n times, we
Page 43 and 44: The standard normalization for the
Page 45 and 46: We assume for simplicity that the i
Page 47 and 48: where w = − t2/3v . 31/3 It follo
Page 49 and 50: 3.5.1 Multiple integrals Propositio
Page 51: When λ > 0, we can deform the inte
Page 54 and 55: plex C, and the complex breaks down
Page 56 and 57: with the substrate is balanced by t
Page 58 and 59: For more information about enzyme k
Page 60 and 61: Thus, the solution involves two ter
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(a) if a(x) > 0, then the boundary
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The inner solution near x = −1 is
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If the fluid interface is a graph z
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where the prime denotes a derivativ
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We seek an asymptotic expansion of
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It follows that dψ dX = −U0 sec
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Chapter 5 Method of Multiple Scales
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Using this expansion in the equatio
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where the overline denotes the aver
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5.3 The method of averaging Conside
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where f : R n ×T → R n is a Lips
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5.5 The WKB method for ODEs Suppose
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Thus, the amplitude of the oscillat
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Asymptotic Analysis and Singular Perturbation Theory

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