Spectral Theory notes 1

Spectral Theory notes 1 - S PECTRAL THEORY OF ORDINARY...

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SPECTRAL THEORY OF ORDINARY DIFFERENTIAL OPERATORS Earl A. Coddin~ton i. Introduction. This is a report on some work which was completed during the last several years~ together with some results which were obtained jointly with A. Dijksma during the 197~-1974 academic year. The work of the author was supported in part by the National Science Foundation under NSF Grant GP-~56~X. The classical eigenvalue problem can be exemplified by the problem on 0 < x < 1 given by (1.1) Lf = Xf, af(O) + bf(1) = 0, where L denotes the formal operator L = id/dx. In case the boundary condition is f(0) - f(1) = 0 we know that there are orthonormal eigenfunetions Xn(X) = ...~ with eigenvalues kn = 2nn~ and there is the exp(-2~ inx)~ n = 0~ ~ !~ eigenvalue expansion ~01 f = r (f, ×n)Xn' (f, ~n ) ~ f~\ " For each f e ~2(0, i) this series converges to f in the metric of ~2(0, i). This is an example of a selfadjoint problem, and, in fact, all problems of the form (i.i) with la I = Ibl ~ 0 are selfadjoint ones. Since % = ~2(0~ i) is a Hilbert spaee~ and since we shall be concerned with spectral theory in Hiibert spaces~ let us interpret the problem (!.i) in the context of this Hilbert space. The f trivially satisfying the boundary condition in (I.i) are those such that f(O) = f(1) = 0. Let S O be the minimal operator in ~ for L. Thus the domain ~(S0) of S O is the set of all those f e such that f is absolutely continuous on 0 < x < i~ f' e 9~ and f(0) = f(1) = 0~ and for f e ~(S0) we have S0f = Lf = if'. This S O is a symmetric operator in ~, (S0f; g) = (f~ S0g ) for all f~ g 6 ~(S0). The maximal operator ~ L is SO, where the graph ~(S0) S O is defined by
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@(S:) = [[h, k] c ~2 = ~ ® ~l(Sof ' h) = (f, k), all f ~ ~(SO) ] . This is the graph of an operator (single-valued function), and S0h = Lh for h c ~(S0) , where ~(S~) is the set of all f ~ ~ which are absolutely continu- ous on 0 < x < 1 and such that f' e ~. For two fixed complex numbers a, b satisfying lal = Ibl ~ 0 define ~(H) = [f ~ ~(S o) laf(O) + bf(1) = 0], where ~b,c ~ C (the complex numbers), and 9 c 9, i]~ll ~ O. trivially satisfying this are the f c ~(So) such that 01 f~ = ~) = . 0 This leads immediately to the consideration of a restriction S of S O with domain ~(S) given by ~(s) = ~(s o) n [~}~, ± where [~] is the subspace spanned by ~ in ~, and [~] = ~ e [~} is the orthogonal complement of [9] ~. This S is symmetric in 9, and we can seek to determine those selfadjoint H S ~H. By analogy with our first example we would expect S c H c S . But what is S ? Identifying it with its graph we would want it to be As before the f 9@ S o ) L. and for f e ~(H) let Hf = Lf. Then S O ~H cS O , in the sense that f~(So) c f~(H) c•(S0) , H is a selfadjoint operator in ~, i.e., H = H . Moreover, all selfadjoint extensions of S O (or selfadjoint restrictions of are of this form. We now seek to broaden the type of problems for the differential operator example, we could consider a side condition for f c ~(S0) For of the form fo af(0) + bf(1) + c f~ = 0,
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(1.2) {{h, k} ~ ~2[(Sf, h) = (f, k), am f ~ ~(s)}. We note that [h, S~h}, h e ~(So) ,* belongs to this set. However, [0, ~} does also, and thus this set is not the graph of an operator in ~, although it is a perfectly nice closed linear manifold in D2 We define S* to be the set (1.2), and it is not difficult to see that s = [[h, Soh + d~}ih ~ ~(So), d ~ C] Since S is not the graph of an operator in
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Spectral Theory notes 1 - S PECTRAL THEORY OF ORDINARY...

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