Non-accretive Schr¨ odinger operators and exponential decay of their eigenfunctions Petr Siegl Mathematical Institute, University of Bern, Switzerland http://gemma.ujf.cas.cz/˜siegl/ Based on [1] D. Krejˇ ciˇ r´ ık, N. Raymond, J. Royer, and P. Siegl: Non-accretive Schr¨ odinger operators and exponential decay of their eigenfunctions Israel Journal of Mathematics, to appear arXiv:1605.02437
Schr¨ odinger operators with complex potentials Main object • Dirichlet realization of L = ( − i ∇ + A ) 2 + V in L 2 (Ω) • Ω ⊂ R d open (no additional assumptions) • V ∈ C 1 (Ω; C ) and A ∈ C 2 (Ω; R d )
Schr¨ odinger operators with complex potentials Main object • Dirichlet realization of L = ( − i ∇ + A ) 2 + V in L 2 (Ω) • Ω ⊂ R d open (no additional assumptions) • V ∈ C 1 (Ω; C ) and A ∈ C 2 (Ω; R d ) • restriction on the growth, oscillations and negative Re V : � � 3 2 + 1 |∇ V ( x ) | + |∇ B ( x ) | = o ( | V ( x ) | + | B ( x ) | ) , � � (Re V ( x )) − = o | V ( x ) | + | B ( x ) | + 1 , | x | → ∞ • where B = ( B jk ) j,k ∈{ 1 ,...,d } , B jk := ∂ j A k − ∂ k A j
Schr¨ odinger operators with complex potentials Main object • Dirichlet realization of L = ( − i ∇ + A ) 2 + V in L 2 (Ω) • Ω ⊂ R d open (no additional assumptions) • V ∈ C 1 (Ω; C ) and A ∈ C 2 (Ω; R d ) • restriction on the growth, oscillations and negative Re V : � � 3 2 + 1 |∇ V ( x ) | + |∇ B ( x ) | = o ( | V ( x ) | + | B ( x ) | ) , � � (Re V ( x )) − = o | V ( x ) | + | B ( x ) | + 1 , | x | → ∞ • where B = ( B jk ) j,k ∈{ 1 ,...,d } , B jk := ∂ j A k − ∂ k A j Objectives 1. find the Dirichlet realization with ρ ( L ) � = ∅ and describe Dom( L ) 2. prove the exponential decay of eigenfunctions of L (due to Im V and B )
Why complex potentials? • superconductivity 1 i ∂ y − x 2 � 2 + i y , − ∂ 2 x + � L 2 ( R 2 ) in • optics with gains and losses 2 − ∆ + (1 + i xy ) e − x 2 e − y 2 , L 2 ( R 2 ) in • hydrodynamics 3 − d 2 i d x 2 + x 2 + L 2 ( R ) εf ( x ) , in • open systems 4 , quantum resonances 5 , damped wave equation 6 ,. . . 1 Y. Almog, B. Helffer, and X.-B. Pan. Trans. Amer. Math. Soc. (2013), pp. 1183–1217. 2 A. Regensburger et al. Phys. Rev. Lett. 107 (2011), p. 233902; J. Yang. Opt. Lett. 39 (2014), pp. 1133–1136. 3 I. Gallagher, T. Gallay, and F. Nier. Int. Math. Res. Not. IMRN (2009), pp. 2147–2199. 4 P. Exner. Open quantum systems and Feynman integrals. D. Reidel Publishing Co., 1985. 5 A. A. Abramov, A. Aslanyan, and E. B. Davies. J. Phys. A: Math. Gen. 34 (2001), p. 57. 6 J. Sj¨ ostrand. Publ. Res. Inst. Math. Sci. 36 (2000), pp. 573–611.
Towards the Dirichlet realization: form methods Simple 1D examples in L 2 ( R ) − d 2 − d 2 − d 2 d x 2 − e x 2 + i e x 4 d x 2 − x 2 + i x 3 , d x 2 + i x 3 ,
Towards the Dirichlet realization: form methods Simple 1D examples in L 2 ( R ) − d 2 − d 2 − d 2 d x 2 − e x 2 + i e x 4 d x 2 − x 2 + i x 3 , d x 2 + i x 3 , Lax-Milgram theorem Let 1. ( V , �· , ·� V ) be a Hilbert space continuously embedded and dense in H 2. ℓ : V × V → C be a continuous sesquilinear form 3. ℓ be V -elliptic ( V -coercive or coercive ) ∃ δ > 0 , ∀ f ∈ V , | ℓ ( f, f ) | ≥ δ � f � V
Towards the Dirichlet realization: form methods Simple 1D examples in L 2 ( R ) − d 2 − d 2 − d 2 d x 2 − e x 2 + i e x 4 d x 2 − x 2 + i x 3 , d x 2 + i x 3 , Lax-Milgram theorem Let 1. ( V , �· , ·� V ) be a Hilbert space continuously embedded and dense in H 2. ℓ : V × V → C be a continuous sesquilinear form 3. ℓ be V -elliptic ( V -coercive or coercive ) ∃ δ > 0 , ∀ f ∈ V , | ℓ ( f, f ) | ≥ δ � f � V Then the (densely defined) operator L Dom( L ) = { f ∈ V : ∃ g ∈ H , ∀ v ∈ V , ℓ ( f, v ) = � g, v �} , L f = g is bijective from Dom( L ) onto H (= ⇒ ρ ( L ) � = ∅ ).
Towards the Dirichlet realization Assumptions Lax-Milgram theorem 1. ( V , �· , ·� V ) be a Hilbert space continuously embedded and dense in H 2. ℓ : V × V → C be a continuous sesquilinear form 3. ℓ be V -elliptic ( V -coercive or coercive ) ∃ δ > 0 , ∀ f ∈ V , | ℓ ( f, f ) | ≥ δ � f � V Why it doesn’t work?
Towards the Dirichlet realization Assumptions Lax-Milgram theorem 1. ( V , �· , ·� V ) be a Hilbert space continuously embedded and dense in H 2. ℓ : V × V → C be a continuous sesquilinear form 3. ℓ be V -elliptic ( V -coercive or coercive ) ∃ δ > 0 , ∀ f ∈ V , | ℓ ( f, f ) | ≥ δ � f � V Why it doesn’t work? • natural candidate for the form of − ∂ 2 x + i x 3 : � ℓ ( f, f ) = �− f ′′ + i x 3 f, f � = � f ′ � 2 + i x 3 | f ( x ) | 2 d x, R
Towards the Dirichlet realization Assumptions Lax-Milgram theorem 1. ( V , �· , ·� V ) be a Hilbert space continuously embedded and dense in H 2. ℓ : V × V → C be a continuous sesquilinear form 3. ℓ be V -elliptic ( V -coercive or coercive ) ∃ δ > 0 , ∀ f ∈ V , | ℓ ( f, f ) | ≥ δ � f � V Why it doesn’t work? • natural candidate for the form of − ∂ 2 x + i x 3 : � ℓ ( f, f ) = �− f ′′ + i x 3 f, f � = � f ′ � 2 + i x 3 | f ( x ) | 2 d x, R • variational space (form domain) 3 3 V = Dom( ℓ ) = H 1 ( R ) ∩ Dom( | x | � · � 2 V = � · � 2 2 · � 2 2 ) , H 1 + �| x |
Towards the Dirichlet realization Assumptions Lax-Milgram theorem 1. ( V , �· , ·� V ) be a Hilbert space continuously embedded and dense in H 2. ℓ : V × V → C be a continuous sesquilinear form 3. ℓ be V -elliptic ( V -coercive or coercive ) ∃ δ > 0 , ∀ f ∈ V , | ℓ ( f, f ) | ≥ δ � f � V Why it doesn’t work? • natural candidate for the form of − ∂ 2 x + i x 3 : � ℓ ( f, f ) = �− f ′′ + i x 3 f, f � = � f ′ � 2 + i x 3 | f ( x ) | 2 d x, R • variational space (form domain) 3 3 V = Dom( ℓ ) = H 1 ( R ) ∩ Dom( | x | � · � 2 V = � · � 2 2 · � 2 2 ) , H 1 + �| x | !!! no coercivity
Towards the Dirichlet realization: new Lax-Milgram Generalized Lax-Milgram theorem of Almog-Helffer 7 Let 1. ( V , �· , ·� V ) be a Hilbert space continuously embedded and dense in H 2. ℓ : V × V → C be a continuous sesquilinear form 7 Y. Almog and B. Helffer. Commun. Part. Diff. Eq. 40 (2015), pp. 1441–1466. 8 A. S. Bonnet-Ben Dhia, P. Ciarlet Jr., and C. M. Zw¨ olf. J. Comput. Appl. Math. 234 (2010), pp. 1912–1919; A. F. M. ter Elst, M. Sauter, and H. Vogt. J. Funct. Anal. 269 (2015), pp. 705–744; L. Grubiˇ si´ c et al. Mathematika 59 (2013), pp. 169–189.
Towards the Dirichlet realization: new Lax-Milgram Generalized Lax-Milgram theorem of Almog-Helffer 7 Let 1. ( V , �· , ·� V ) be a Hilbert space continuously embedded and dense in H 2. ℓ : V × V → C be a continuous sesquilinear form 3. A-H coercivity: ∃ Φ 1 , Φ 2 bounded linear maps on V and H and | ℓ ( f, f ) | + | ℓ (Φ 1 f, f ) | ≥ δ � f � 2 V , ∃ δ > 0 , ∀ f ∈ V , | ℓ ( f, f ) | + | ℓ ( f, Φ 2 f ) | ≥ δ � f � 2 V . 7 Y. Almog and B. Helffer. Commun. Part. Diff. Eq. 40 (2015), pp. 1441–1466. 8 A. S. Bonnet-Ben Dhia, P. Ciarlet Jr., and C. M. Zw¨ olf. J. Comput. Appl. Math. 234 (2010), pp. 1912–1919; A. F. M. ter Elst, M. Sauter, and H. Vogt. J. Funct. Anal. 269 (2015), pp. 705–744; L. Grubiˇ si´ c et al. Mathematika 59 (2013), pp. 169–189.
Towards the Dirichlet realization: new Lax-Milgram Generalized Lax-Milgram theorem of Almog-Helffer 7 Let 1. ( V , �· , ·� V ) be a Hilbert space continuously embedded and dense in H 2. ℓ : V × V → C be a continuous sesquilinear form 3. A-H coercivity: ∃ Φ 1 , Φ 2 bounded linear maps on V and H and | ℓ ( f, f ) | + | ℓ (Φ 1 f, f ) | ≥ δ � f � 2 V , ∃ δ > 0 , ∀ f ∈ V , | ℓ ( f, f ) | + | ℓ ( f, Φ 2 f ) | ≥ δ � f � 2 V . Then the (densely defined) operator L Dom( L ) = { f ∈ V : ∃ g ∈ H , ∀ v ∈ V , ℓ ( f, v ) = � g, v �} , L f = g is bijective from Dom( L ) onto H (= ⇒ ρ ( L ) � = ∅ ). 7 Y. Almog and B. Helffer. Commun. Part. Diff. Eq. 40 (2015), pp. 1441–1466. 8 A. S. Bonnet-Ben Dhia, P. Ciarlet Jr., and C. M. Zw¨ olf. J. Comput. Appl. Math. 234 (2010), pp. 1912–1919; A. F. M. ter Elst, M. Sauter, and H. Vogt. J. Funct. Anal. 269 (2015), pp. 705–744; L. Grubiˇ si´ c et al. Mathematika 59 (2013), pp. 169–189.
Towards the Dirichlet realization: new Lax-Milgram Generalized Lax-Milgram theorem of Almog-Helffer 7 Let 1. ( V , �· , ·� V ) be a Hilbert space continuously embedded and dense in H 2. ℓ : V × V → C be a continuous sesquilinear form 3. A-H coercivity: ∃ Φ 1 , Φ 2 bounded linear maps on V and H and | ℓ ( f, f ) | + | ℓ (Φ 1 f, f ) | ≥ δ � f � 2 V , ∃ δ > 0 , ∀ f ∈ V , | ℓ ( f, f ) | + | ℓ ( f, Φ 2 f ) | ≥ δ � f � 2 V . Then the (densely defined) operator L Dom( L ) = { f ∈ V : ∃ g ∈ H , ∀ v ∈ V , ℓ ( f, v ) = � g, v �} , L f = g is bijective from Dom( L ) onto H (= ⇒ ρ ( L ) � = ∅ ). • similar recent results 8 7 Y. Almog and B. Helffer. Commun. Part. Diff. Eq. 40 (2015), pp. 1441–1466. 8 A. S. Bonnet-Ben Dhia, P. Ciarlet Jr., and C. M. Zw¨ olf. J. Comput. Appl. Math. 234 (2010), pp. 1912–1919; A. F. M. ter Elst, M. Sauter, and H. Vogt. J. Funct. Anal. 269 (2015), pp. 705–744; L. Grubiˇ si´ c et al. Mathematika 59 (2013), pp. 169–189.
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