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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 18.102 Introduction to Functional Analysis Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . 140 LECTURE NOTES FOR 18.102, SPRING 2009 Lecture 24. Thursday, May 7: Completeness of Hermite basis Here is what I claim was done last time. Starting from the ground state for the harmonic oscillator d 2 (24.1) H = − dx 2 + x 2 , Hu = u , u = e − x 2 / 2 and using the creation and annihilation operators d d (24.2) A = + x, C = − + x, AC − CA = 2Id , H = CA + Id dx dx I examined the higher eigenfunctions: (24.3) u j = C j u = p j ( x ) u ( c ) , p ( x ) = 2 j x j + l.o.ts , Hu j = (2 j + 1) u j and showed that these are orthogonal, u j ⊥ u k , j = k, and so when normalized give an orthonormal system in L 2 ( R ) : u j (24.4) e j = 2 j/ 2 ( j !) 1 1 . 2 π 4 Now, what I want to show today, and not much more, is that the e j form an orthonormal basis of L 2 ( R ) , meaning they are complete as an orthonormal sequence. There are various proofs of this, but the only ‘simple’ ones I know involve the Fourier inversion formula and I want to use the completeness to prove the Fourier inversion formula, so that will not do. Instead I want to use a version of Mehler’s formula. I also tried to motivate this a bit last time. Namely, I suggested that to show the completeness of the e j ’s it is enough to find a compact selfadjoint operator with these as eigenfunctions and no null space. It is the last part which is tricky. The first part is easy. Remembering that all the e j are real, we can find an operator with the e j ;s as eigenfunctions with corresponding eigenvalues λ j > (say) by just defining ∞ ∞ (24.5) Au ( x ) = λ j ( u,e j ) e j ( x ) = λ j e j ( x ) e j ( y ) u ( y ) ....
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This note was uploaded on 11/08/2011 for the course PHY 18.102 taught by Professor Staff during the Spring '09 term at MIT.
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