Five Lectures : Psychoanalysis Politics and Utopia by Herbert Marcuse

By Herbert Marcuse

Psychoanalysis, Politics, Philosophy, Humanities

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The indicial equation is s 2 = 0, with s = 0 as a root of multiplicity 2, independently of c. One solution is given by a power series in r , while another involves log r . We discard the solution with the logarithm because it would represent a singularity at the middle of the drum. To get at the sign of c, we use the condition R(1) = 0 and argue as follows: Without loss of generality, R(0) is positive. Suppose c > 0, and let r1 ≤ 1 be the first value of r > 0 where R(r1 ) = 0. From the equation r −1 (r R ) = c R and the inequality R(r ) > 0 for 0 < r < r1 , we see that r R is strictly increasing for 0 < r < r1 .

The equation for R(r ) becomes r 2 R + r R + (cr 2 − n 2 )R = 0. This has a regular singular point at r = 0, and the indicial equation is s 2 = n 2 . Thus s = ±n. In fact, we can recognize this equation as Bessel’s equation of order √ n by a change of variables: A little argument excludes c ≤ 0. Putting k = c, ρ = kr , and y(ρ) = R(r ) leads to y + ρ −1 y + (1 − n 2 ρ −2 )y = 0, which is exactly Bessel’s equation of order n. Transforming the solution y(ρ) = Jn (ρ) back with r = k −1 ρ, we see that R(r ) = y(ρ) = Jn (ρ) = Jn (kr ) is a solution of the equation for R.

Let λ be in E, and let ϕ be a nonzero solution of (SL) corresponding to λ and normalized so that ϕ r = 1. Multiplying (SL1) by ϕ¯ and integrating, we have b λ= b λ|ϕ|2r dt = − a = − pϕ ϕ¯ a b a b + a b ( pϕ ) ϕ¯ dt + q|ϕ|2 dt a b p|ϕ |2 dt + q|ϕ|2 dt a b ≥ − p(b)ϕ (b)ϕ(b) + p(a)ϕ (a)ϕ(a) + (|ϕ|2r )(r −1 q) dt a ≥ − p(b)ϕ (b)ϕ(b) + p(a)ϕ (a)ϕ(a) + inf {r (t)−1 q(t)}. a≤t≤b Let us show under the hypotheses c1 c2 ≤ 0 and d1 d2 ≥ 0 that ϕ (a)ϕ(a) ≥ 0 and ϕ (b)ϕ(b) ≤ 0, and then the asserted lower bounds will follow.

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