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Nevanlinna–Herglotz Functions and Some Applications to Spectral Theory

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Nevanlinna–Herglotz Functions and Some Applications to Spectral Theory Nevanlinna{Herglotz Functions and Some Applications to Spectral Theory Fritz Gesztesy (Baylor University, Waco, TX, USA) Herglotz-Nevanlinna Functions and Their Applications Institut Mittag-Leffler, Djursholm, Sweden May 8{12, 2017 Fritz Gesztesy (Baylor Univ.) Nevanlinna{Herglotz Functions May 9, 2017 1 / 96 Outline 1 Topics Discussed 2 Scalar Nevanlinna{Herglotz Functions 3 Matrix-Valued Nevanlinna{Herglotz Functions 4 Operator-Valued Nevanlinna{Herglotz Functions 5 Epilogue Fritz Gesztesy (Baylor Univ.) Nevanlinna{Herglotz Functions May 9, 2017 2 / 96 Topics Discussed Topics Discussed: • Scalar Nevanlinna{Herglotz Functions − Basic facts − Plenty of Examples (from very elementary to more sophisticated ones) − More basic facts − Aronszajn{Donoghue and Simon{Wolff theory − A model approach to rank-one perturbations − A model approach to self-adjoint extensions of symmetric operators with deficiency indices (1; 1) − The model approach applies to continuous and discrete half-line Hamiltonian systems such as, Sturm{Liouville operators (up to 3 coefficients), Dirac-type operators, Jacobi operators (tri-diagonal), CMV (Cantero{Moral{Vel´azquez) operators (five-diagonal, but with lots of zeros strategically placed make it effectively a 2nd order operator) − Literature hints Fritz Gesztesy (Baylor Univ.) Nevanlinna{Herglotz Functions May 9, 2017 3 / 96 Topics Discussed Topics Discussed (contd.): • Matrix-Valued Nevanlinna{Herglotz Functions − Basic facts − Matrix analogs of Aronszajn{Donoghue and Simon{Wolff theory − Applies to continuous and discrete half-line and full-line Hamiltonian systems − Literature hints • Operator-Valued Nevanlinna{Herglotz Functions − Basic facts − Applies to Schr¨odinger,Dirac-type, and Jacobi operators with operator-valued coefficients − Applies to Dirichlet-to-Neumann (resp., Robin-to-Robin) operators for elliptic PDEs − Literature hints Fritz Gesztesy (Baylor Univ.) Nevanlinna{Herglotz Functions May 9, 2017 4 / 96 Topics Discussed Topics Discussed (contd.): • Epilogue − Hints at some of the questions asked during the talk and subsequently during the conference are provided together with a few more references. Fritz Gesztesy (Baylor Univ.) Nevanlinna{Herglotz Functions May 9, 2017 5 / 96 Scalar Nevanlinna{Herglotz Functions Scalar Nevanlinna{Herglotz Functions: Basics Let C± = fz 2 C j ±Im(z) > 0g. Definition 1 m : C+ ! C is called a Nevanlinna{Herglotz function (in short, a N-H fct.) if m is analytic on C+ and m(C+) ⊆ C+. That is, we're looking at analytic self-maps on C+. But actually, it suffices to assume m(C+) ⊆ C+. Unless explicitly stated otherwise, we always extend m to C− by reflection, i.e., m(z) = m(z); z 2 C−: Fritz Gesztesy (Baylor Univ.) Nevanlinna{Herglotz Functions May 9, 2017 6 / 96 Scalar Nevanlinna{Herglotz Functions Scalar Nevanlinna{Herglotz Functions: Basics (contd.) There is considerable disagreement concerning the proper name of functions satisfying the conditions in Definition 1: One finds the names Nevanlinna, Pick, Nevanlinna-Pick, Herglotz, and R-functions. Sometimes this depends on the geographical origin of authors and at times whether the open upper half-plane C+ or the conformally equivalent open unit disk D is involved). However, Herglotz definitely studied what's also called Caratheodory functions (functions analytic in the open unit disk with nonnegative real part), and Nevanlinna studied the open upper half-plane, C+. So pure mathematicians got this right by calling them Nevanlinna functions, and mathematical physicists, who settled on Herglotz functions, definitely got this wrong. Anway, an object with so many names must be really important! Fritz Gesztesy (Baylor Univ.) Nevanlinna{Herglotz Functions May 9, 2017 7 / 96 Scalar Nevanlinna{Herglotz Functions Scalar Nevanlinna{Herglotz Functions: Basics (contd.) Gustav Herglotz (2 February 1881 { 22 March 1953): According to Wikipedia, a German Bohemian math- ematician, best known for his works on the theory of relativity and seismology, also worked in many areas of applied and pure mathematics. (E.g., celestial me- chanics, theory of electrons, special and general rela- tivity, hydrodynamics, differential geometry, number theory.)Habilitation under Felix Klein in G¨ottingen, 1904. After detours via Vienna and Leipzig he be- came the successor of Carl Runge in G¨ottingenin 1925. One of his students was Emil Artin. Fritz Gesztesy (Baylor Univ.) Nevanlinna{Herglotz Functions May 9, 2017 8 / 96 Scalar Nevanlinna{Herglotz Functions Scalar Nevanlinna{Herglotz Functions: Basics (contd.) Rolf Herman Nevanlinna (22 October 1895 { 28 May 1980): According to Wikipedia, one of the most famous Finnish mathematicians, particularly appreciated for his work in complex analysis. His most important mathematical achievement is the value distribution theory of meromorphic functions. Rolf Nevanlinna's article \Zur Theorie der meromorphen Funktionen", which contains the Main Theorems was published in Acta Mathematica in 1925. Hermann Weyl called it\one of the few great mathematical events of the century". Fritz Gesztesy (Baylor Univ.) Nevanlinna{Herglotz Functions May 9, 2017 9 / 96 Scalar Nevanlinna{Herglotz Functions Scalar Nevanlinna{Herglotz Fcts: Basics (contd.) Addition and Composition: If m(z) and n(z) are Nevanlinna{Herglotz functions, then so are m(z) + n(z) and m(n(z)). Elementary examples of Nevanlinna{Herglotz functions are, e.g., c + id; c + dz; c 2 R; d ≥ 0; zr ; 0 < r < 1; ln(z); choosing the obvious branches, and tan(z); − cot(z); a2;1 + a2;2z a1;1 a1;2 2×2 ∗ 0 −1 ; a = 2 C ; a j2a = j2; j2 = ; a1;1 + a1;2z a2;1 a2;2 1 0 i.e., certain linear fractional transformations (the group of automorphisms of C+), hence the special case −1=z. Fritz Gesztesy (Baylor Univ.) Nevanlinna{Herglotz Functions May 9, 2017 10 / 96 Scalar Nevanlinna{Herglotz Functions Scalar Nevanlinna{Herglotz Fcts: Basics (contd.) As a consequence, − 1=m(z); m(−1=z); ln(m(z)); and a2;1 + a2;2m(z) ma(z) = ; a1;1 + a1;2m(z) with a 2 C2×2 as above, are all N-H functions whenever m(z) is N-H. Most importantly, let H be a self-adjoint operator in a separable complex Hilbert space H with (·; ·)H the scalar product on H × H linear in the second factor. Consider the resolvent of H,(H − z)−1, z 2 CnR. Then for all f 2 H, −1 (f ; (H − z) f )H; z 2 CnR; is the prime example of a scalar N-H function (use the spectral theorem). Actually, −1 (H − z) ; z 2 CnR; is the prime example of an operator-valued N-H function. Fritz Gesztesy (Baylor Univ.) Nevanlinna{Herglotz Functions May 9, 2017 11 / 96 Scalar Nevanlinna{Herglotz Functions Scalar Nevanlinna{Herglotz Fcts: Basics (contd.) The fundamental result on N-H functions, in part due to Fatou, Herglotz, Luzin, Nevanlinna, Plessner, Privalov, de la Vall´eePoussin, Riesz, and others: Theorem 2 Let m(z) be a Nevanlinna{Herglotz function. Then (i) m(z) has finite normal limits m(λ ± i0) = lim"#0 m(λ ± i") for a.e. λ 2 R. (ii) Suppose m(z) has a zero normal limit on a subset of R having positive Lebesgue measure. Then m ≡ 0. (iii) The Nevanlinna (resp., Riesz{Herglotz) representation holds: Z 1 λ m(z)= c + dz + d!(λ) − ; z 2 ; λ − z 1 + λ2 C+ R c = Re(m(i)); d = lim m(iη)=(iη) ≥ 0; η"1 Z d!(λ) 1 < 1: I.e., \Superpositions" of ! prime example .... 1 + λ2 λ − z R Fritz Gesztesy (Baylor Univ.) Nevanlinna{Herglotz Functions May 9, 2017 12 / 96 Scalar Nevanlinna{Herglotz Functions Scalar Nevanlinna{Herglotz Fcts: Basics (contd.) Theorem 2 (contd.) (iv) Let (λ1; λ2) ⊂ R, then the Stieltjes inversion formula for ! reads Z λ2 1 1 −1 ! (fλ1g)+ ! (fλ2g)+ !((λ1; λ2))= π lim dλ Im(m(λ + i")): "#0 2 2 λ1 (v) The absolutely continuous (ac) part !ac of ! with respect to Lebesgue measure dλ on R is given by −1 d!ac (λ) = π Im(m(λ + i0))dλ: (vi) Any poles and isolated zeros of m are simple and located on the real axis, the residues at poles being negative. Actually, much more is true: Denote by ! = !ac + !s = !ac + !sc + !pp the decomposition of ! into its absolutely continuous (ac), singularly continuous (sc), pure point (pp), and singular (s) parts with respect to Lebesgue measure on R. Fritz Gesztesy (Baylor Univ.) Nevanlinna{Herglotz Functions May 9, 2017 13 / 96 Scalar Nevanlinna{Herglotz Functions Scalar Nevanlinna{Herglotz Fcts: Basics (contd.) Theorem 3 Let m(z) be a Nevanlinna{Herglotz function with representation as above. Then (i) d = 0 and R d!(λ)(1 + jλjs )−1 < 1 for some s 2 (0; 2) R R 1 −s if and only if 1 dη η Im(m(iη)) < 1. (ii) Let (λ1; λ2) ⊂ R, η1 > 0. Then there is a constant C(λ1; λ2; η1) > 0 such that "jm(λ + i")j ≤ C(λ1; λ2; η1), (λ, ") 2 [λ1; λ2] × (0;"0). (iii) sup ηjm(iη)j < 1 if and only if m(z) = R d!(λ)(λ − z)−1 η>0 R and R d!(λ) < 1. R In this case, R d!(λ) = sup ηjm(iη)j = −i limη"1 ηm(iη). R η>0 (iv) For all λ 2 R, lim"#0 "Re(m(λ + i")) = 0, !(fλg) = lim"#0 "Im(m(λ + i")) = −i lim"#0 "m(λ + i"). Fritz Gesztesy (Baylor Univ.) Nevanlinna{Herglotz Functions May 9, 2017 14 / 96 Scalar Nevanlinna{Herglotz Functions Scalar Nevanlinna{Herglotz Fcts: Basics (contd.) Theorem 3 (contd.) (v) Let L > 0 and suppose 0 ≤ Im(m(z)) ≤ L for all z 2 C+. Then d = 0, ! is purely absolutely continuous, ! = !ac , and d!(λ) −1 −1 0 ≤ = π lim Im(m(λ + i")) ≤ π L for a.e. λ 2 R: dλ "#0 (vi) Let p 2 (1; 1),[λ3; λ4] ⊂ (λ1; λ2), [λ1; λ2] ⊂ (λ5; λ6). If Z λ2 sup dλ jIm(m(λ + i"))jp < 1; (∗) 0<"<1 λ1 then ! = !ac is purely absolutely continuous on (λ1; λ2), d!ac p −1 d!ac p dλ 2 L ((λ1; λ2); dλ), and lim"#0 kπ Im(m(· + i")) − dλ kL ((λ3,λ4);dλ) = 0.
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