Spectral Shift Functions and Dirichlet-to-Neumann Maps Fritz Gesztesy (Baylor University, Waco, TX, USA) Based on various joint collaborations with J. Berndt (TU-Graz, Austria), S. Clark (Missouri S & T, Rolla, MO, USA), K. A. Makarov (Univ. of Missouri, Columbia, MO, USA), S. N. Naboko (St. Petersburg State Univ, Russia), S. Nakamura (Univ. of Tokyo, Japan), R. Nichols (UTC, TN, USA), and M. Zinchenko (UNM, Albuquerque, NM, USA) IWOTA 2017, Technical University of Chemnitz, Germany August 14 – 18, 2017 Fritz Gesztesy (Baylor University, Waco) SSF and D-N Maps Chemnitz, August 17, 2017 1 / 38
Outline Topics discussed 1 Notation 2 1d Schr¨ odinger Operators on a Finite Interval 3 4 Boundary Data Maps for 1d Schr¨ odinger Operators SSF, Boundary Triples, Abstract Weyl–Titchmarsh Fcts. 5 Applications to PDEs 6 Fritz Gesztesy (Baylor University, Waco) SSF and D-N Maps Chemnitz, August 17, 2017 2 / 38
Topics discussed Topics discussed: • A warm up: Self-adjoint extensions , Krein-type resolvent formulas for 1 d Schr¨ odinger operators • Resolvent trace formulas. • Krein–Lifshitz spectral shift (SSF) functions. • Hints at an extension of SSF , the Spectral Shift Operator (SSO) , whose trace equals SSF . • Connect SSO with abstract Weyl–Titchmarsh M -operators . • Sketch applications of Dirichlet-to-Neumann maps , more generally, abstract Weyl–Titchmarsh M -operators , to PDEs . Fritz Gesztesy (Baylor University, Waco) SSF and D-N Maps Chemnitz, August 17, 2017 3 / 38
Topics discussed Some Literature: In the 1d context: F. G. and M. Zinchenko, Symmetrized perturbation determinants and applications to boundary data maps and Krein-type resolvent formulas , Proc. London Math. Soc. (3) 104 , 577–612 (2012). S. Clark, F.G., R. Nichols, and M. Zinchenko, Boundary data maps and Krein’s resolvent formula for Sturm–Liouville operators on a finite interval , Operators and Matrices 8 , 1–71 (2014). In the Abstract and PDE context: F.G., K. A. Makarov, and S. N. Naboko, The spectral shift operator , in Mathematical Results in Quantum Mechanics , J. Dittrich, P. Exner, and M. Tater (eds.), Operator Theory: Advances and Applications, Vol. 108, Birkh¨ auser, Basel, 1999, pp. 59–90. J. Behrndt, F.G., and S. Nakamura , Spectral shift functions and Dirichlet- -to-Neumann maps , arXiv:1609.08292, submitted to Math. Ann. Fritz Gesztesy (Baylor University, Waco) SSF and D-N Maps Chemnitz, August 17, 2017 4 / 38
Notation A Bit of Notation: H denotes a (separable, complex ) Hilbert space, I H represents the identity operator in H . If A is a closed (typically, self-adjoint) operator in H , then ρ ( A ) ⊆ C denotes the resolvent set of A ; z ∈ ρ ( A ) ⇐ ⇒ A − z I H is a bijection. σ ( A ) = C \ ρ ( A ) denotes the spectrum of A . σ p ( A ) denotes the point spectrum (i.e., the set of eigenvalues) of A . σ d ( A ) denotes the discrete spectrum of A (i.e., isolated eigenvalues of finite (algebraic) multiplicity). If A is closable in H , then A denotes the operator closure of A in H . Note. All operators will be linear in the following. Fritz Gesztesy (Baylor University, Waco) SSF and D-N Maps Chemnitz, August 17, 2017 5 / 38
Notation A Bit of Notation (contd.): If A is closable in H , then A denotes the operator closure of A in H . B ( H ) is the set of bounded operators defined on H . B p ( H ), 1 ≤ p ≤ ∞ denotes the p th trace ideal of B ( H ), � ( T ∗ T ) 1 / 2 � p < ∞ , where J ⊆ N is an ⇒ � (i.e., T ∈ B p ( H ) ⇐ j ∈J λ j appropriate index set, and the eigenvalues λ j ( T ) of T are repeated according to their algebraic multiplicity), B 1 ( H ) is the set of trace class operators, B 2 ( H ) is the set of Hilbert–Schmidt operators, B ∞ ( H ) is the set of compact operators. tr H ( A ) = � j ∈J λ j ( A ) denotes the trace of A ∈ B 1 ( H ). Fritz Gesztesy (Baylor University, Waco) SSF and D-N Maps Chemnitz, August 17, 2017 6 / 38
1d Schr¨ odinger Operators on a Finite Interval Maximal and Minimal Schr¨ odinger Operators in 1d We’ll use the 1d case of Schr¨ odinger operators as a warm up case: Let V ∈ L 1 ((0 , R ); dx ) be real-valued , R ∈ (0 , ∞ ) , and introduce the Schr¨ odinger differential expression τ via τ = − d 2 dx 2 + V ( x ) , x ∈ (0 , R ) , and the associated maximal and minimal operators in L 2 ((0 , R ); dx ) associated with τ by H max f = τ f , � � � � g , g ′ ∈ AC([0 , R ]); τ g ∈ L 2 ((0 , R ); dx ) g ∈ L 2 ((0 , R ); dx ) f ∈ dom( H max ) = , H min f = τ f , f ∈ dom( H min ) = { g ∈ dom( H max ) | g (0) = g ′ (0) = g ( R ) = g ′ ( R ) = 0 } . AC([0 , R ]) denotes the set of absolutely continuous functions on [0 , R ]. Fritz Gesztesy (Baylor University, Waco) SSF and D-N Maps Chemnitz, August 17, 2017 7 / 38
1d Schr¨ odinger Operators on a Finite Interval Self-Adjoint Extensions of H min H min H min Introduce the following families of self-adjoint extensions H θ 0 ,θ R and H K ,φ in L 2 ((0 , R ); dx ) of the minimal operator H min , H θ 0 ,θ R f = τ f , θ 0 , θ R ∈ [0 , π ) , separated b.c.’s, � � � cos( θ 0 ) g (0) + sin( θ 0 ) g ′ (0) = 0 , f ∈ dom( H θ 0 ,θ R ) = g ∈ dom( H max ) � cos( θ R ) g ( R ) − sin( θ R ) g ′ ( R ) = 0 and H K ,φ f = τ f , φ ∈ [0 , 2 π ) , K ∈ SL(2 , R ) , coupled b.c.’s, � � � g ( R ) � � g (0) � � � � = e i φ K f ∈ dom( H K ,φ ) = g ∈ dom( H max ) . � g ′ ( R ) g ′ (0) SL(2 , R ) denotes the set of 2 × 2 matrices with determinant = 1 and real entries. Claim: There’s nothing else that’s self-adjoint! Fritz Gesztesy (Baylor University, Waco) SSF and D-N Maps Chemnitz, August 17, 2017 8 / 38
1d Schr¨ odinger Operators on a Finite Interval Self-Adjoint Extensions of H min H min H min (contd.) Indeed, one can unify separated and coupled boundary conditions as follows: Theorem. The operator H F , G , � � � � � �� � g (0) g ( R ) � H F , G f = τ f , f ∈ dom( H F , G ) = g ∈ dom( H max ) � F = G , g ′ (0) g ′ ( R ) is a self-adjoint extension of H min if and only if there exist matrices F , G ∈ C 2 × 2 � 0 − 1 � G ) = 2, FJF ∗ = GJG ∗ , J = satisfying rank( F . 1 0 In particular, the case of separated boundary conditions corresponds to � cos( θ 0 ) � � � sin( θ 0 ) 0 0 θ 0 , θ R ∈ [0 , π ) . F = , G = , 0 0 − cos ( θ R ) sin( θ R ) The case of coupled (i.e., non-separated ) boundary conditions corresponds to F = e i φ K , G = I 2 , K ∈ SL(2 , R ) , φ ∈ [0 , 2 π ) . Fritz Gesztesy (Baylor University, Waco) SSF and D-N Maps Chemnitz, August 17, 2017 9 / 38
Boundary Data Maps for 1d Schr¨ odinger Operators The Basics of Boundary Data Maps Boundary Data Maps: Define the boundary trace map, γ F , G , associated with the boundary { 0 , R } of (0 , R ) and the 2 × 2 parameter matrices F , G satisfying rank( F G ) = 2, � 0 − 1 � FJF ∗ = GJG ∗ , J = , by 1 0 C 1 ([0 , R ]) → C 2 , � � � � γ F , G : u (0) u ( R ) u �→ F − G . u ′ (0) u ′ ( R ) Then, � � � � F 1 , 1 − G 1 , 1 F 1 , 2 G 1 , 2 γ F , G = D F , G γ D + N F , G γ N , D F , G = , N F , G = , F 2 , 1 − G 2 , 1 F 2 , 2 G 2 , 2 where γ D and γ N represent Dirichlet and Neumann traces , � u (0) � � − u ′ (0) � γ D u = , γ N u = . u ′ ( R ) u ( R ) Moreover, define S F ′ , G ′ , F , G = N F ′ , G ′ D ∗ F , G − D F ′ , G ′ N ∗ F , G . Fritz Gesztesy (Baylor University, Waco) SSF and D-N Maps Chemnitz, August 17, 2017 10 / 38
Boundary Data Maps for 1d Schr¨ odinger Operators The Basics of Boundary Data Maps (contd.) Let F , G ∈ C 2 × 2 be such that rank( F G ) = 2, and assume that z ∈ ρ ( H F , G ). Then the boundary value problem � � c 1 − u ′′ + Vu = zu , u , u ′ ∈ AC ([0 , R ]) , ∈ C 2 , γ F , G u = c 2 has a unique solution u ( z , · ) = u F , G ( z , · ; c 1 , c 2 ) for each c 1 , c 2 ∈ C . Let F , G , F ′ , G ′ ∈ C 2 × 2 with F , G satisfying rank( F G ) = 2, FJF ∗ = GJG ∗ , � 0 − 1 � , and similarly for F ′ , G ′ . Assuming z ∈ ρ ( H F , G ), we introduce the J = 1 0 boundary data map (an extension of Dirichlet-to Neumann and Robin-to-Robin maps ) by F , G ( z ) : C 2 → C 2 , Λ F ′ , G ′ � � c 1 Λ F ′ , G ′ = Λ F ′ , G ′ F , G ( z ) F , G ( z ) γ F , G u F , G ( z , · ; c 1 , c 2 ) c 2 = γ F ′ , G ′ u F , G ( z , · ; c 1 , c 2 ) , where u F , G ( z , · ; c 1 , c 2 ) satisfies the above boundary value problem. Fritz Gesztesy (Baylor University, Waco) SSF and D-N Maps Chemnitz, August 17, 2017 11 / 38
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