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This is Green’s Second Identity: pr:G2Id1 b S, L0 u − L0 S, u = τ (x) u(x) a d ∗ d S (x) − S ∗ (x) u(x) . 5) In the literature, the expressions for the Green’s identities take τ = −1 eq2G2Id and V = 0 in the operator L0 . 1. I. C. 18) can simplify Green’s 2nd Identity. If S and u correspond to physical quantities, they must satisfy RBC. We will verify this statement for two special cases: the closed string and the open string. 13) u(a, t) = u(b, t), S ∗ (a, t) = S ∗ (b, t), 16 CHAPTER 2. GREEN’S IDENTITIES d ∗ S dx τ (a) = τ (b), = x=a d ∗ S dx d d u = u .

The Green’s identities are: (a) Green’s first identity: b S, L0 u = − S ∗ (x)τ (x) a b + dx a d u(x) dx d ∗ d S τ (x) u(x) dx dx + S ∗ (x)V (x)u(x) , (b) Green’s second identity: b S, L0 u − L0 S, u = τ (x) u(x) a d ∗ d S (x) − S ∗ (x) u(x) . dx dx 2. , RBC) the linear operator L0 is Hermitian: S, L0 u = u, L0 S ∗ . see FW p207 expl. 6. 6 21 References Green’s formula is described in [Stakgold67, p70] and [Stakgold79, p167]. The derivation of the potential energy of a string was inspired by [Simon71,p390].

And boundary condition [ˆ nS · ∇ + κS ][a1 u1 (x) + a2 u2 (x)] = a1 [ˆ nS · ∇ + κS ]u1 + a2 [ˆ nS · ∇ + κS ]u2 = a1 (0) + a2 (0) = 0. We have thus shown that L0 [a1 u1 + a2 u2 ] = a1 L0 u1 + a2 L0 u2 . 4) This is called the principle of superposition, and it is the defining property of a linear operator. 26: [L0 − ω 2 σ(x)]u(x) = σ(x)f (x). pr:DeltaFn1 The delta function is defined by the equation pr:Fcd d Fcd = eq3deltdef pr:Fcd1 c dxδ(x − xk ) = 1 if c < xk < d 0 otherwise. 5) where Fcd represents the total force over the interval [c, d].

### Analysis of the Stability of Astrophysical Objects and a New Mechanism for Freeing Their Energy

by Paul

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