Math212a1419 the Discrete and the Essential Spectrum, Compact and Relatively Compact Operators, the Schr¨Odingeroperator

Home , Discrete spectrum, Essential spectrum

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 → → ∞

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. → ∞ → ∞ 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.

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 − { } spectrum of H will be denoted by σd (H) or simply by σd when H is fixed in the discussion.

The discrete spectrum and the essential spectrum.

Characterizing the discrete spectrum.

If λ σd (H) then for sufficiently small > 0 the spectral ∈ 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.

The converse.

Conversely, suppose that λ σ(H) and that P(λ , λ + ) is ∈ − 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 { | − }

L2(S(λ−,λ+), dµ)

is ﬁnite dimensional.

L2(S(λ−,λ+), dµ) is finite dimensional. If we write [ S = (λ , λ + ) n (λ−,λ+) − × { } n∈N then since Mˆ L2(S(λ−,λ+)) = L2((λ , λ + ), n ) n − { } 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. − × { }

For all but ﬁnitely many n we have

µ ((λ , λ + ) n ) = 0. − × { } 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.

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.

We conclude from this lemma that there are at most ﬁnitely many points (sr , k) with sr (λ , λ + ) which have ﬁnite measure in ∈ − 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 ∈ have proved Proposition.

λ σ(H) belongs to σd (H) if and only if there is some > 0 such ∈ that P((λ , λ + ))(H) is ﬁnite dimensional. −

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 ∈ 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 ψ1, . . . , ψn is an orthonormal basis of Ran(K) and ψ is any element of H then

n X Kψ = (Kψ, ψj )ψj . j=1

∗ If we set φj := K ψj we can write this last equation as

n X Kψ = (ψ, φj )ψj . 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, ∈ Ran(K), is ﬁnite dimensional, in which case the dimension of its range is called the rank of K.

Finite rank operators.

An operator K L(H) is said to be of ﬁnite rank if its range, ∈ Ran(K), is ﬁnite dimensional, in which case the dimension of its range is called the rank of K. If ψ1, . . . , ψn is an orthonormal basis of Ran(K) and ψ is any element of H then

n X Kψ = (Kψ, ψj )ψj . j=1

∗ If we set φj := K ψj we can write this last equation as

n X Kψ = (ψ, φj )ψj . j=1

∗ The elements φj = K ψj are linearly independent, for if P aj φj = 0 then X X X ( aj ψj , Kf ) = (aj ψj , (f , φk )ψk ) = (f , aj φj ) = 0 j,k j

for all f H which can not happen unless all the aj = 0 since ∈ ψ1, . . . , ψn is an orthonormal basis of Ran(K). So

n X Kψ = (ψ, φj )ψj j=1

is the most general expression of a ﬁnite rank operator.

Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schrödingeroperator. If K is an operator of finite rank and B is a bounded operator then clearly BK and KB are of finite rank.

Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl’s theorem on the essential spectrum.

n X Kψ = (ψ, φj )ψj j=1 is the most general expression of a ﬁnite rank operator. Then K ∗ is given by

∗ X K f = (f , ψj )φj

as can immediately be checked, and so the adjoint of a ﬁnite rank operator is of ﬁnite rank.

n X Kψ = (ψ, φj )ψj j=1 is the most general expression of a ﬁnite rank operator. Then K ∗ is given by

∗ X K f = (f , ψj )φj

as can immediately be checked, and so the adjoint of a finite rank operator is of finite rank. If K is an operator of finite rank and B is a bounded operator then clearly BK and KB are of finite rank.

Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schr¨odingeroperator. So K = Kn on the space spanned by the ﬁrst n φj ’s. Let Hn be the orthocomplement of this space, i.e. the space spanned by φn+1, φn+2,... . So

K Kn = sup Kψ . k − k ψ∈Hn,kψk=1 k k

Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl’s theorem on the essential spectrum.

A compact operator is the norm limit of ﬁnite rank operators.

Proof. Let K be a compact operator and φj an orthonormal Pn { } basis of H. Let Kn := ( , φj )Kφj . j=1 ·

A compact operator is the norm limit of ﬁnite rank operators.

Proof. Let K be a compact operator and φj an orthonormal Pn { } basis of H. Let Kn := ( , φj )Kφj . So K = Kn on the space j=1 · spanned by the ﬁrst n φj ’s. Let Hn be the orthocomplement of this space, i.e. the space spanned by φn+1, φn+2,... . So

K Kn = sup Kψ . k − k ψ∈Hn,kψk=1 k k

The numbers K Kn are decreasing and hence tend to some k − k limit, `, and so we can ﬁnd a sequence ψn with ψn Hn, ψn = 1 ∈ k k and lim Kψn `. The ψn converge weakly to 0, and hence so k k ≥ do the Kψn. We can choose a (strongly) convergent subsequence which must therefore converge to 0, and so ` = 0.

The converse.

Over the next few slides we will assume that K is the norm limit of finite rank operators. Theorem Suppose that K is the norm limit of finite rank operators and is self-adjoint. Then σess(K) 0 with equality if and only if H is ⊂ { } infinite dimensional. We must show that if λ = 0 then for 0 < < λ the image of the 6 | | spectral projection PK (λ , λ + ) is finite dimensional. Without − loss of generality we may assume that λ > 0.

Proof.

Let Kn be a sequence of finite rank operators such that K Kn 1/n. If the image of the spectral projection k − k ≤ PK (λ , λ + ) is infinite dimensional, we can find ψn in this − image with ψn = 1 and Knψn = 0. Then k k 1 ((K Kn)ψn, ψn) = (Kψn, ψn) λ , n ≥ | − | | | ≥ − a contradiction for large enough n.

A normal form for norm limits of ﬁnite rank operators. Singular values

Theorem Suppose that K is the norm limit of ﬁnite rank operators. There exist orthonormal sets ψj and φj and positive numbers { } { } sj = sj (K) (called the singular values of K) such that

X ∗ X K = sj ( , φj )ψj , K = sj ( , ψj )φj . · · j j

There are either ﬁnitely many sj (if K is of ﬁnite rank) or they converge to 0.

Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schr¨odingeroperator. If ζ is orthogonal to all the φj then we know by the spectral theorem that K ∗Kζ ζ for any > 0 and hence K ∗Kζ = 0 k k ≤ k k which implies that Kζ = 0. We can now proceed to the proof of the theorem:

Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl’s theorem on the essential spectrum.

A normal form for norm limits of ﬁnite rank operators.

We know that K ∗K is the norm limit of ﬁnite rank operators and is self-adjoint so we can apply the preceding theorem. We also know that K ∗Kψ = 0 (ψ, K ∗Kψ) = Kψ 2 = 0 so ⇒ k k ker(K ∗K) = ker K.

2 ∗ Let sj be the non-zero eigenvalues of K K (arranged in decreasing order) and φj a corresponding orthonormal set.

A normal form for norm limits of ﬁnite rank operators.

2 ∗ Let sj be the non-zero eigenvalues of K K (arranged in decreasing order) and φj a corresponding orthonormal set.

If ζ is orthogonal to all the φj then we know by the spectral theorem that K ∗Kζ ζ for any > 0 and hence K ∗Kζ = 0 k k ≤ k k which implies that Kζ = 0. We can now proceed to the proof of the theorem:

A normal form for norm limits of ﬁnite rank operators.

Proof. For any ψ H we have ∈ X ∗ ψ = (ψ, φj )φj + ζ, ζ ker(K K) = ker(K). ∈ j

−1 P Let ψj := sj Kφj . Then Kψ = j sj (ψ, φj )ψj . To see that the ψj are orthonormal, observe that

−1 −1 −1 −1 ∗ (ψj , ψk ) = sj sk (Kφj , Kφk ) = sj sk (K Kφj , φk )

−1 = sj sk (φj , φk ). The formula for K ∗ follows by taking adjoints.

A normal form for norm limits of ﬁnite rank operators.

P From K = j sj ( , φj )ψj we see that Kφj = sj ψj and from the ∗ · ∗ formula for K we see that K ψj = sj φj . Hence

∗ 2 ∗ 2 K Kφj = sj φj and KK ψj = sj ψj .

Characterizations of compact operators.

Let K be a bounded operator on H. Theorem The following statements are equivalent: 1 K is the norm limit of ﬁnite rank operators.

2 If An L(H) with An A strongly then AnK AK in norm. ∈ → → 3 ψn ψ weakly implies that Kψn Kψ in norm. → → 4 K is compact.

We have already proved that 4. implies 1.

Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schr¨odingeroperator. PN Write K as the ﬁnite sum K = sj ( , φj )ψj . Then j=1 · N N 2 X 2 2 X 2 AnK sup sj (ψ, φj ) Anψj sj Anψj 0. k k ≤ | | k k ≤ k k → kψk=1 1 1

Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl’s theorem on the essential spectrum.

1. 2. ⇒ Replacing An by An A we may assume that A = 0. By the − uniform boundedness principle, there is some M such that An M. If Kj is a sequence of finite rank operators with k k ≤ Kj K in norm, then → AnK An(K Kj ) + AnKj M K Kj + AnKj so we k k ≤ k − k k k ≤ k − k k k may assume in proving 1. 2. that K is of finite rank. ⇒ PN Write K as the finite sum K = sj ( , φj )ψj . Then j=1 · N N 2 X 2 2 X 2 AnK sup sj (ψ, φj ) Anψj sj Anψj 0. k k ≤ | | k k ≤ k k → kψk=1 1 1

2. 3. ⇒ Again replacing ψn by ψn ψ we may assume that ψn 0 weakly. − → Choose any φ with φ = 1 and consider the rank one operators k k An := ( , ψn)φ. Then An 0 strongly. Hence, by 2., · → ∗ ∗ AnK = sup (K ζ, ψn) φ 0. k k kζk=1 | |k k →

But ∗ sup (K ζ, ψn) = Kψn kζk=1 | | k k

so Kψn 0. k k →

3. 4. ⇒ If ψn is bounded, it has a weakly convergent subsequence. (Just choose an orthonomal basis φj and then a subsequence that all the

(ψnk , φj ) converge.) Then apply 3. to this subsequence. This completes the proof of the theorem.

Hilbert Schmidt integral operators.

Hilbert-Schmidt operators aka operators given by L2 integral kernels.

Let (M, µ) be a measure space and K = K(x, y) L2(M M, µ µ). For ψ L2(M, µ) deﬁne Kψ by ∈ × Z ⊗ ∈ Kψ(x) := K(x, y)ψ(y)dµ(y). M So by Cauchy-Schwarz, Z Z Z 2 2 Kψ(x) dµ(x) = K(x, y)ψ(y)dµ(y) dµ(x) M | | M M Z Z Z K(x, y) 2dµ(y) ψ(y) 2dµ(y) dµ(x) = ≤ M M | | M | | Z Z K(x, y) 2dµ(x)dµ(y) ψ(y) 2dµ(y) . M×M | | M | | 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.

Hilbert Schmidt integral operators.

Z Z Z Kψ(x) 2 K(x, y) 2dµ(x)dµ(y) ψ(y) 2dµ(y) . M | | ≤ M×M | | M | |

We see that K is a bounded operator on L2(M, µ).

Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schr¨odingeroperator. We can expand K in terms of this orthonomal basis: X Z K(x, y) = ci,j φi (x)φj (y), ci,j = K(x, y)φi (x)φj (y)dµ(x)dµ(y) i,j

so ci,j = (Kφj , φi ), and X Kψ(x) = ci,j (ψ, φj )φi (x). i,j

Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl’s theorem on the essential spectrum.

Hilbert Schmidt integral operators. Hilbert Schmidt operators are compact.

Let φj be an orthonormal basis of L2(M, µ) so that φi (x)φj (y) is an orthonormal basis of L2(M M, µ µ). × ⊗

Hilbert Schmidt integral operators. Hilbert Schmidt operators are compact.

Let φj be an orthonormal basis of L2(M, µ) so that φi (x)φj (y) is an orthonormal basis of L2(M M, µ µ). We can expand K in × ⊗ terms of this orthonomal basis: X Z K(x, y) = ci,j φi (x)φj (y), ci,j = K(x, y)φi (x)φj (y)dµ(x)dµ(y) i,j

so ci,j = (Kφj , φi ), and X Kψ(x) = ci,j (ψ, φj )φi (x). i,j

Hilbert Schmidt integral operators.

X Kψ(x) = ci,j (ψ, φj )φi (x). i,j P 2 Since ci,j < we see that K can be approximated in norm i,j | | ∞ by ﬁnite rank operators and hence is compact.

Hilbert Schmidt integral operators. When is a compact operator Hilbert-Schmidt?

Let H = L2(M, µ) and K a compact operator on H with sj its { } singular values. Proposition. K is Hilbert Schmidt if and only if X s2 < j ∞ j

in which case K has an integral kernel with

X Z s2 = K(x, y) 2dµ(x)dµ(y). j | | j M

Hilbert Schmidt integral operators. Proof. P We know that K = sj ( , φj )ψj , for orthonormal sets φj and ψj . j · Replacing this (possibly inﬁnite) sum by a ﬁnite sum gives

N X KN := sj ( , φj )ψj · j=1

as an approximating ﬁnite rank operator to K. Now KN has the PN integral kernel KN (x, y) = j=1 sj φj (y)ψj (x). R 2 PN 2 Furthermore KN (x, y) dµ(x)dµ(y) = s . If one M×M | | j=1 j side of X Z s2 = K(x, y) 2dµ(x)dµ(y) j | | j M converges, so does the other.

Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schr¨odingeroperator. Since every Hilbert space is isomorphic to some L2(M, µ) we see that the Hilbert Schmidt operators together with the norm

1   2 X 2 K 2 =  sj (K)  k k j

form a Hilbert space isomorphic to L2(M M, µ µ). × ⊗

Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl’s theorem on the essential spectrum.

Hilbert Schmidt operators in general. A more general deﬁnition of Hilbert Schmidt operator.

Let us call a compact operator on a Hilbert space H Hilbert P 2 Schmidt if sj (K) < . In case H = L2(M, µ) we know that j ∞ this coincides with our earlier deﬁnition.

Hilbert Schmidt operators in general. A more general deﬁnition of Hilbert Schmidt operator.

Let us call a compact operator on a Hilbert space H Hilbert P 2 Schmidt if sj (K) < . In case H = L2(M, µ) we know that j ∞ this coincides with our earlier deﬁnition.

Since every Hilbert space is isomorphic to some L2(M, µ) we see that the Hilbert Schmidt operators together with the norm

1   2 X 2 K 2 =  sj (K)  k k j

form a Hilbert space isomorphic to L2(M M, µ µ). × ⊗

Hilbert Schmidt operators in general. Another characterization of Hilbert-Schmidt operators.

Proposition. A compact operator K is Hilbert Schmidt if and only if there is an orthonormal basis ζn with { } X 2 Kζn < n k k ∞ in which case this holds for any orthonormal basis. Furthermore

2 X 2 K 2 = Kζn . k k n k k

Hilbert Schmidt operators in general.

Proof. P 2 P 2 P ∗ 2 We have Kζn = (Kζn, ψj ) = ζn, K ψj ) = n k k n,j | | n,j | | X ∗ 2 X 2 = K ψn = sj (K) . k k n j

All terms in the sums are positive so they converge or diverge together and this holds for any orthonormal basis.

Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schr¨odingeroperator. For a function f we will let f (p) denote the operator

ψ −1 (f (p)( ψ)(p)) . 7→ F F In other words, we take the Fourier transform ψˆ = (ψ), then F multiply by f (p) and then inverse Fourier transform back.

Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl’s theorem on the essential spectrum.

Hilbert Schmidt operators in general. An important example.

n Let H = L2(R ) and let denote the Fourier transform (extended F to H via Plancherel). For any function g we will let g(x) (bad notation) denote the operator of multiplication by g on H (where deﬁned).

Hilbert Schmidt operators in general. An important example.

ψ −1 (f (p)( ψ)(p)) . 7→ F F In other words, we take the Fourier transform ψˆ = (ψ), then F multiply by f (p) and then inverse Fourier transform back.

Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schr¨odingeroperator. If f and g are n n n in L2(R ) then this kernel belongs to L2(R R ) and so is × Hilbert Schmidt. By symmetry, the operator f (p)g(x) is then also Hilbert Schmidt.

Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl’s theorem on the essential spectrum.

Hilbert Schmidt operators in general.

So, for example, the operator g(x)f (p) is given by

1 Z ψ ˇ ψ (g(x)f (p) )(x) = n/2 g(x)f (x y) (y)dy (2π) Rn − where fˇ denotes the inverse Fourier transform of f .

Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schr¨odingeroperator. By symmetry, the operator f (p)g(x) is then also Hilbert Schmidt.

Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl’s theorem on the essential spectrum.

Hilbert Schmidt operators in general.

So, for example, the operator g(x)f (p) is given by

1 Z ψ ˇ ψ (g(x)f (p) )(x) = n/2 g(x)f (x y) (y)dy (2π) Rn − where fˇ denotes the inverse Fourier transform of f . If f and g are n n n in L2(R ) then this kernel belongs to L2(R R ) and so is × Hilbert Schmidt.

Hilbert Schmidt operators in general.

So, for example, the operator g(x)f (p) is given by

Hilbert Schmidt operators in general.

Theorem Let f and g be bounded (Borel measurable) functions which tend to 0 at . Then f (p)g(x) and g(x)f (p) are compact. ∞

Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schr¨odingeroperator. We have

g(x)f (p) gR (x)fR (p) g ∞ f fR ∞ + g gR ∞ fR ∞, k − k ≤ k k k − k k − k k k where the norm on the left is the operator norm.

So gR (x)fR (p) approaches g(x)f (p) in norm.

Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl’s theorem on the essential spectrum.

Hilbert Schmidt operators in general.

Proof.

By symmetry it is enough to consider g(x)f (p). Let fR be the function equal to f on the ball of radius R centered at the origin and zero outside, with a similar deﬁnition of gR . Then gR and fR belong to L2 and hence gR (x)fR (p) is Hilbert Schmidt and hence compact.

Hilbert Schmidt operators in general.

Proof.

g(x)f (p) gR (x)fR (p) g ∞ f fR ∞ + g gR ∞ fR ∞, k − k ≤ k k k − k k − k k k where the norm on the left is the operator norm.

So gR (x)fR (p) approaches g(x)f (p) in norm.

Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schr¨odingeroperator. This result will be of importance to us in conjunction with Weyl’s theorem on the essential spectrum.

Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl’s theorem on the essential spectrum.

Hilbert Schmidt operators in general.

For example, suppose that H0 is the free Hamiltonian and we consider its resolvent at some negative real number, for example at 1. This operator is of the form f (p) where − 1 f (p) = . −1 + p 2 | | Let V be a function which vanishes at . Then the operator ∞ V (x)f (p) is compact.

Hilbert Schmidt operators in general.

Weyl’s characterization of the essential spectrum.

Recall the following from the lecture on the square well: Proposition.

A λ R belongs to the essential spectrum, σess (A), of a self ∈ adjoint operator A, if and only if there exists a sequence ψn D(A) such that ∈ 1 ψn = 1, k k 2 ψn converges weakly to 0, and

3 (λI A)ψn 0. − → Such a sequence is called a singular Weyl sequence. I want to quickly remind you how we proved this Proposition:

Characterizing the spectrum as “approximate eigenvalues”.

Theorem Let A be a self-adjoint operator on a Hilbert space H. A real number λ belongs to the spectrum of A if and only if there exists a sequence of vectors un D(A) such that ∈

un = 1 and (λI A)un 0. k k k − k →

If λ σ(A) then (λI A)−1 exists and is bounded, so there is a 6∈ − µ > 0 such that (λI A)u µ u u D(A). So if k − k ≥ k k ∀ ∈ un = 1 then (λI A)un µ and can not approach 0. k k k − k ≥ So if a sequence as in the theorem exists, then λ belongs to the spectrum of A. This is the easy direction; it is practically the deﬁnition of the spectrum. We now turn to the other direction: 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.

Suppose that no such sequence exists. I claim that there exists a constant c > 0 such that

u c (λI A)u u D(A). (1) k k ≤ k − k ∀ ∈

For if not, there would exist a sequence vn of non-zero elements of D(A) with (λI A)vn / vn 0. k − k k k → Replacing vn by un := vn/ vn gives a sequence of unit vector in k k D(A) with λI A)un 0, proving (1). k − k → From the inequality (1) we conclude that the map λI A is − injective. We can also conclude that its image is closed. For if wn = (λI A)un, un D(A) with wn w, then (1) implies that − ∈ → the sequence un is Cauchy and so converges to some u H. We ∈ want to show that u D(A). For any v D(A) we have: ∈ ∈ 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.

(u, (λI A)v) = lim (un, (λI A)v) = lim((λI A)un, v) − n→∞ − − = lim (wn, v) = (w, v). n→∞

So u D((λI A)∗) = D(A∗) = D(A) since A is self-adjoint. To ∈ − show that λ is in the resolvent set of A we must show that the image of (λI A) is all of H: −

Let f H and consider the linear function on the image of λI A ∈ − given by w (v, f ) where w = (λI A)v. 7→ − Now (v, f ) f v and by (1), this is c f w . So this linear | ≤ k kk k ≤ k k k function is bounded on the image of λI A. But this image, being − a closed subspace of H as we have just proved, is a Hilbert space in its own right, and so we may apply the Riesz representation theorem to conclude that there is a u in the image of λI A such − that ((λI A)v, u) = (v, f ) v D(A). − ∀ ∈ Since A, and hence λI A is self-adjoint, this implies that − u D(A) and (λI A)u = f . ∈ − We now review Weyl’s theorem characterizing the essential spectrum:

Suppose that λ belongs to the discrete spectrum of A. Let Hλ denote the eigenspace with eigenvalue λ so Hλ is ﬁnite dimensional by assumption. Decompose H into the direct sum

⊥ H = Hλ H . ⊕ λ The entire interval (λ , λ + ) lies in the resolvent set of the − restriction of A to H⊥. So the restriction of λI A to H⊥ has a λ − λ bounded inverse. So if ψn D(A) is a sequence of elements of H ∈ with

ψn = 1 and k k (λI A)ψn 0 − → ⊥ then the Hλ components of the ψn must tend to 0. The Hλ components then form a bounded sequence in a ﬁnite dimensional space, and hence we can extract a convergent subsequence. So we have proved one half of the following theorem of Weyl: Theorem A point λ belongs to the essential spectrum of a self-adjoint operator A if and only if there exists a sequence ψn D(A) such ∈ that

ψn = 1, k k ψn has no convergent subsequence, and

(λI A)ψn 0. − →

Conversely, suppose that λ lies in the essential spectrum. We want to construct a sequence as in the theorem. If λ is an eigenvalue of infinite multiplicity, we can construct an orthonormal sequence of eigenvectors ψn so (λI A)ψn = 0 and the ψn have no convergent − subsequence. Otherwise we can find a sequence of λn lying in the spectrum of A with λn = λ and λn λ. So λ λn = 0 and λ λn lies in the 6 → − 6 − spectrum of A λnI and hence by our theorem characterizing the − spectrum as “approximate eigenvalues”, we can find ψn with ψn = 1 and k k 1 (λnI A)ψn λn λ , k − k ≤ n| − | so (λI A)ψn 0. We wish to prove that ψn has no convergent − → subsequence. If it did, then passing to the subsequence (and relabeling) we would have ψn ψ with ψ = 1 and ψ an → k k eigenvector of A with eigenvalue λ. Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schrödingeroperator. Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl’s theorem on the essential spectrum.

Then

(λ λn)(ψn, ψ) = (ψn, Aψ) λn(ψn, ψ) = ((A λI )ψn, ψ), − − − so 1 λ λn (ψn, ψ) λn λ , | − || | ≤ n| − | which is impossible since (ψn, ψ) 1. →

We now turn to the proof of the proposition: A sequence ψn is said to converge weakly to ψ if for every v H ∈

(ψn, v) (ψ, v). → It is easy to check that every bounded sequence in a separable Hilbert space has a weakly convergent subsequence. On the other hand, suppose that ψn H satisﬁes ∈

ψn = 1 and ψn converges weakly to 0. k k

Then ψn can have no (strongly) convergent subsequence because the strong limit of any subsequence would have to equal the weak limit and hence = 0 contradicting the hypothesis ψn = 1. k k I repeat the statement of the proposition we want to prove:

Proposition λ R belongs to the essential spectrum of a self adjoint operator ∈ A if and only if there exists a sequence ψn D(A) such that ∈ 1 ψn = 1, k k 2 ψn converges weakly to 0, and

3 (λI A)ψn 0. − →

If the first two conditions are satisfied then the ψn can not have a convergent subsequence, as we have just seen. So the conditions of Weyl’s theorem on the essential specrum that we have just proved are satisfied, and hence λ belongs to the essential spectrum of A.

Conversely, if λ belongs to the essential spectrum of A then we can ﬁnd a sequence ψn satisfying the three conditions of the above theorem. The ﬁrst condition implies that the ψn are uniformly bounded so we can choose a subsequence which converges weakly to some ψ H. Let us pass to this subsequence (and relabel). ∈ Since our ψn has no convergent subsequence, there is an > 0 such that ψn ψ k − k ≥ for all n. By the third condition in the theorem,

(ψn, (A λI )φ) = ((A λI )ψn, φ) 0 φ D(A) − − → ∀ ∈ so (ψ, (A λI )φ) = 0 φ D(A). − ∀ ∈ Since A is self-adjoint, this implies that ψ D(A) and Aψ = λψ. ∈ 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.

Consider the sequence 1 ψ˜n := (ψn ψ). ψn ψ − k − k

We have ψñ = 1 and (ψñ, u) 0 for all u H. So the first two k k → ∈ conditions of the Proposition are satisfied. So is the third because 1 (A λI )ψñ = (A λI )ψn 0 − ψn ψ − → k − k since ψn ψ > 0. k − k ≥

Weyl’s theorem.

Let A and B be self-adjoint operators. Let z be a point in the resolvent set of A and of B and let RA = RA(z) and RB = RB (z) be the corresponding resolvents. Theorem

If RA RB is compact, then −

σess (A) = σess (B).

Before proving the theorem observe the following identity involving the resolvent: 1 1 RA(z)(λI A) = RA(z)(zI A + (λ z)I ) = z λ − z λ − − − − 1 = I RA(z). z λ − − So if ψn is a singular Weyl sequence at λ then 1 RA(z) I ψn 0. − z λ → −

Proof.

Since the ψn converge weakly to 0 and RA RB is compact, we − know that (RA RB )ψn converges strongly to zero and hence − 1 RB (z) I ψn 0. − z λ → − 1 In particular RB (z)ψn = 0. But k k → |z−λ| 6 1 1 RB (z) I = (λI B)RB (z) − z λ λ z − − − so (after an innocuous renormalizing) the RB (z)ψn are a singular Weyl sequence for B. So σess (A) σess (B). Interchanging the ⊂ roles of A and B then proves the theorem.

Recall: The second resolvent identity.

Let a and b be operators whose range is the whole space and with bounded inverses. Then

a−1 b−1 = a−1(b a)b−1 − − assuming that the right hand side is deﬁned. For example, if A and B are closed operators with D(B A) D(A) we get − ⊃

RA(z) RB (z) = RA(z)(B A)RB (z). − − This is known as the second resolvent identity.

Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schr¨odingeroperator. From the second resolvent identity we see that if K is relatively compact with respect to to a self adjoint operator A then RA+K (z) RA(z) is compact. Therefore: −

Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl’s theorem on the essential spectrum.

Let us now say that an operator K is relatively compact with respect to to an operator A if KRA(z) is compact for some z in the resolvent set of A. Notice that from the ﬁrst resolvent identity,

RA(z) RA(w) = (z w)RA(z)RA(w), − −

we see that if KRA(z) is compact for some z then KRA(z) is compact for all z in the resolvent set. So the condition is independent of z.

Let us now say that an operator K is relatively compact with respect to to an operator A if KRA(z) is compact for some z in the resolvent set of A. Notice that from the ﬁrst resolvent identity,

RA(z) RA(w) = (z w)RA(z)RA(w), − −

we see that if KRA(z) is compact for some z then KRA(z) is compact for all z in the resolvent set. So the condition is independent of z. From the second resolvent identity we see that if K is relatively compact with respect to to a self adjoint operator A then RA+K (z) RA(z) is compact. Therefore: −

Weyl’s theorem

Theorem. [Hermann Weyl.] If K is a self-adjoint operator which is relatively compact with respect to to a self adjoint operator A then σess(A + K) = σess(A).

Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schrödingeroperator. The purpose of the next few slides is to study the opposite situation - where the potential tends to infinity at infinity. We will show that under this circumstance H0 + V has no essential spectrum. For this we will need a lemma.

Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl’s theorem on the essential spectrum.

Applications to Schr¨odingeroperators. Applications to Schr¨odingeroperators.

1 The resolvent of H0 at 1 is 2 which tends to 0 as − − 1+kpk p . If V is a bounded potential which tends to zero at k k → ∞ inﬁnity then VR( 1, H0) is − 1 V (x) . − 1 + p 2 k k In view of what we have proved above, we conclude that

σess(H0 + V ) = σess(H0).

Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schr¨odingeroperator. For this we will need a lemma.

Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl’s theorem on the essential spectrum.

Applications to Schr¨odingeroperators. Applications to Schr¨odingeroperators.

σess(H0 + V ) = σess(H0).

The purpose of the next few slides is to study the opposite situation - where the potential tends to infinity at infinity. We will show that under this circumstance H0 + V has no essential spectrum. Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schrödingeroperator. Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl’s theorem on the essential spectrum.

Applications to Schr¨odingeroperators. Applications to Schr¨odingeroperators.

σess(H0 + V ) = σess(H0).

The purpose of the next few slides is to study the opposite situation - where the potential tends to infinity at infinity. We will show that under this circumstance H0 + V has no essential spectrum. For this we will need a lemma. Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schrödingeroperator. Let V 0 and set H = H0 + V . Notice that H 0 as an ≥ 1 ≥ operator, and in fact H H0. So for any u Dom(H 2 ) ≥ ∈ 1 1 2 2 H u = (u, H0u) (u, Hu) = H 2 u k 0 k ≤ k k

1 1 so H 2 is H 2 bounded and since (u, Hu) (u, (H + I )u) we see 0 ≤ that 1 1 2 − 2 (H0 + I )(H + I ) is bounded. More directly, this is obvious in a multiplicative spectral representation.

Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl’s theorem on the essential spectrum.

Applications to Schr¨odingeroperators.

First observe that for all potentials W of compact support, the 1 2 −1 operator W (I + H0 ) is compact. This is again because it is of the form g(x)f (p) where f and g are functions going to zero at inﬁnity. Let V 0 and set H = H0 + V . Notice that H 0 as an ≥ 1 ≥ operator, and in fact H H0. So for any u Dom(H 2 ) ≥ ∈ 1 1 2 2 H u = (u, H0u) (u, Hu) = H 2 u k 0 k ≤ k k

1 1 so H 2 is H 2 bounded and since (u, Hu) (u, (H + I )u) we see 0 ≤ that 1 1 2 − 2 (H0 + I )(H + I ) is bounded. More directly, this is obvious in a multiplicative spectral representation.

Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schr¨odingeroperator. Proof.

1 1 1 1 − 2 2 −1 2 − 2 W (I + H) = W (I + H0 ) (I + H0 )(I + H) and we know that the product of the ﬁrst two factors is compact and the product of the second two factors is bounded. So − 1 W (I + H) 2 is compact. Multiply by the bounded operator − 1 −1 (I + H) 2 on the right to conclude that W (I + H) is compact.

Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl’s theorem on the essential spectrum.

Applications to Schr¨odingeroperators.

Lemma If V 0 is a non-negative potential and W is multiplication by a ≥ bounded function of compact support then W is H = H0 + V compact, i.e. W (I + H)−1 is compact.

Applications to Schr¨odingeroperators.

Lemma If V 0 is a non-negative potential and W is multiplication by a ≥ bounded function of compact support then W is H = H0 + V compact, i.e. W (I + H)−1 is compact.

Proof.

Applications to Schr¨odingeroperators. A theorem of Friedrichs, 1932.

Theorem

If V (x) as x then σess(H0 + V ) is empty. → ∞ → ∞ Proof. For any E write V E = f g where f 0 and g has compact − − ≥ support. See the next slide. By the lemma, g is H0 + f compact, so σess(H0 + f ) = σess(H E). Since f 0, we know that − ≥ σess(H E) [0, ) so σess(H) [E, ). Since this is true for − ⊂ ∞ ⊂ ∞ all E we conclude that σess(H0 + V ) is empty.

Applications to Schr¨odingeroperators.

f + E

E g E −

V = f + E g −

Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schr¨odingeroperator.