Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl's theorem on the essential spectrum. Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schr¨odingeroperator. Shlomo Sternberg November 20, 2014 Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schr¨odingeroperator. Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl's theorem on the essential spectrum. The main results of today's lecture are about the Schr¨odinger operator H = H0 + V . They are: If V is bounded and V 0 as x then ! ! 1 σess(H) = σess(H0): So this is a generalization of the results of the lecture on the square well. I will review some definitions and facts about the discrete and the essential spectrum. If V as x then σess(H) is empty. ! 1 ! 1 Both of these results will require a jazzed up version of Weyl's theorem on the stability of the essential spectrum as worked out in lecture on the square well. This in turn will require some results on compact operators that we did not do in Lecture 3. Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schr¨odingeroperator. Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl's theorem on the essential spectrum. 1 The discrete and the essential spectrum. 2 Finite rank operators. A normal form for norm limits of finite rank operators. 3 Compact operators. 4 Hilbert Schmidt operators Hilbert Schmidt integral operators. Hilbert Schmidt operators in general. 5 Weyl's theorem on the essential spectrum. Applications to Schr¨odingeroperators. Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schr¨odingeroperator. The complement in σd (H) in σ(H) is called the essential spectrum of H and is denoted by σess(H) or simply by σess when H is fixed in the discussion. Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl's theorem on the essential spectrum. The discrete spectrum and the essential spectrum. Let H be a self-adjoint operator on a Hilbert space H and let σ = σ(H) R denote is spectrum. ⊂ Recall that the discrete spectrum of H is defined to be those eigenvalues λ of H which are of finite multiplicity and are also isolated points of the spectrum. This latter condition says that there is some > 0 such that the intersection of the interval (λ , λ + ) with σ consists of the single point λ . The discrete − f g spectrum of H will be denoted by σd (H) or simply by σd when H is fixed in the discussion. Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schr¨odingeroperator. Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl's theorem on the essential spectrum. The discrete spectrum and the essential spectrum. Let H be a self-adjoint operator on a Hilbert space H and let σ = σ(H) R denote is spectrum. ⊂ Recall that the discrete spectrum of H is defined to be those eigenvalues λ of H which are of finite multiplicity and are also isolated points of the spectrum. This latter condition says that there is some > 0 such that the intersection of the interval (λ , λ + ) with σ consists of the single point λ . The discrete − f g spectrum of H will be denoted by σd (H) or simply by σd when H is fixed in the discussion. The complement in σd (H) in σ(H) is called the essential spectrum of H and is denoted by σess(H) or simply by σess when H is fixed in the discussion. Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schr¨odingeroperator. Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl's theorem on the essential spectrum. Characterizing the discrete spectrum. If λ σd (H) then for sufficiently small > 0 the spectral 2 projection P = P((λ , λ + )) has the property that it − commutes with H and the restriction of H to the image of P has only λ in its spectrum and hence P(H) is finite dimensional, since the multiplicity of λ is finite by assumption. Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schr¨odingeroperator. Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl's theorem on the essential spectrum. The converse. Conversely, suppose that λ σ(H) and that P(λ , λ + ) is 2 − finite dimensional. This means that in a multiplication spectral representation of H, the subset S(λ−,λ+) of S = σ N × consisting of all (s; n) λ < s < λ + has the property that f j − g L2(S(λ−,λ+); dµ) is finite dimensional. Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schr¨odingeroperator. Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl's theorem on the essential spectrum. L2(S(λ−,λ+); dµ) is finite dimensional. If we write [ S = (λ , λ + ) n (λ−,λ+) − × f g n2N then since M^ L2(S(λ−,λ+)) = L2((λ , λ + ); n ) n − f g we conclude that all but finitely many of the summands on the right are zero, which implies that for all but finitely many n we have µ ((λ , λ + ) n ) = 0: − × f g Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schr¨odingeroperator. Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl's theorem on the essential spectrum. For all but finitely many n we have µ ((λ , λ + ) n ) = 0: − × f g For each of the finite non-zero summands, we can apply the case N = 1 of the following lemma: Lemma N N Let ν be a measure on R such that L2(R ; ν) is finite dimensional. Then ν is supported on a finite set in the sense that there is some finite set of m distinct points x1;:::; xm each of positive measure and such that the complement of the union of these points has ν measure zero. Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schr¨odingeroperator. Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl's theorem on the essential spectrum. Proof. N Partition R into cubes whose vertices have all coordinates of the form t=2r for a vector t with integer coordinates and with integer r and so that this is a disjoint union. The corresponding decomposition of the L2 spaces shows that only finitely many of these cubes have positive measure, and as we increase r the cubes with positive measure are nested downward, and can not increase N in number beyond n = dim L2(R ; ν). Hence they converge in measure to at most n distinct points each of positive ν measure and the complement of their union has measure zero. Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schr¨odingeroperator. Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl's theorem on the essential spectrum. We conclude from this lemma that there are at most finitely many points (sr ; k) with sr (λ , λ + ) which have finite measure in 2 − the multiplicative spectral representation of H, each giving rise to an eigenvector of H with eigenvalue sr , and the complement of these points has measure zero. This shows that λ σd (H). We 2 have proved Proposition. λ σ(H) belongs to σd (H) if and only if there is some > 0 such 2 that P((λ , λ + ))(H) is finite dimensional. − Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schr¨odingeroperator. Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl's theorem on the essential spectrum. Characterizing the essential spectrum This is simply the contrapositive of the preceding proposition: Proposition. λ σ(H) belongs to σess(H) if and only if for every > 0 the space 2 P((λ , λ + ))(H) − is infinite dimensional. Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schr¨odingeroperator. If 1; : : : ; n is an orthonormal basis of Ran(K) and is any element of H then n X K = (K ; j ) j : j=1 ∗ If we set φj := K j we can write this last equation as n X K = ( ; φj ) j : j=1 Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl's theorem on the essential spectrum. Finite rank operators. An operator K L(H) is said to be of finite rank if its range, 2 Ran(K), is finite dimensional, in which case the dimension of its range is called the rank of K. Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schr¨odingeroperator. Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl's theorem on the essential spectrum. Finite rank operators. An operator K L(H) is said to be of finite rank if its range, 2 Ran(K), is finite dimensional, in which case the dimension of its range is called the rank of K.