Exponential decay estimates for fundamental solutions of Schr¨ odinger-type operators Svitlana Mayboroda, Bruno Poggi University of Minnesota Department of Mathematics AMS Special Session on Regularity of PDEs on Rough Domains Boston, Massachusetts April 21, 2018 1 Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨ odinger operators
The electric Schr¨ odinger operator By an electric Schr¨ odinger operator we mean a second-order linear operator of the form L := − div A ∇ + V, where V is a scalar, real-valued, positive a.e., locally integrable function, and A is a matrix of bounded, measurable complex coefficients satisfying the uniform ellipticity condition n λ | ξ | 2 ≤ ℜ e � A ( x ) ξ, ξ � ≡ ℜ e � A ij ( x ) ξ j ¯ ξ i and � A � L ∞ ( R n ) ≤ Λ , i,j =1 (1) for some λ > 0 , Λ < ∞ and for all ξ ∈ C n , x ∈ R n . • The exponential decay of solutions to the Schr¨ odinger operator in the presence of a positive potential is an important property underpinning foundation of quantum physics. • However, establishing a precise rate of decay for complicated potentials is a challenging open problem to this date. (Landis conjecture) 2 Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨ odinger operators
The electric Schr¨ odinger operator By an electric Schr¨ odinger operator we mean a second-order linear operator of the form L := − div A ∇ + V, where V is a scalar, real-valued, positive a.e., locally integrable function, and A is a matrix of bounded, measurable complex coefficients satisfying the uniform ellipticity condition n λ | ξ | 2 ≤ ℜ e � A ( x ) ξ, ξ � ≡ ℜ e � A ij ( x ) ξ j ¯ ξ i and � A � L ∞ ( R n ) ≤ Λ , i,j =1 (1) for some λ > 0 , Λ < ∞ and for all ξ ∈ C n , x ∈ R n . • The exponential decay of solutions to the Schr¨ odinger operator in the presence of a positive potential is an important property underpinning foundation of quantum physics. • However, establishing a precise rate of decay for complicated potentials is a challenging open problem to this date. (Landis conjecture) 2 Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨ odinger operators
The electric Schr¨ odinger operator By an electric Schr¨ odinger operator we mean a second-order linear operator of the form L := − div A ∇ + V, where V is a scalar, real-valued, positive a.e., locally integrable function, and A is a matrix of bounded, measurable complex coefficients satisfying the uniform ellipticity condition n λ | ξ | 2 ≤ ℜ e � A ( x ) ξ, ξ � ≡ ℜ e � A ij ( x ) ξ j ¯ ξ i and � A � L ∞ ( R n ) ≤ Λ , i,j =1 (1) for some λ > 0 , Λ < ∞ and for all ξ ∈ C n , x ∈ R n . • The exponential decay of solutions to the Schr¨ odinger operator in the presence of a positive potential is an important property underpinning foundation of quantum physics. • However, establishing a precise rate of decay for complicated potentials is a challenging open problem to this date. (Landis conjecture) 2 Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨ odinger operators
Exponential Decay of electric Schr¨ odinger operators • First results expressing upper estimates on the solutions in terms of a certain distance associated to V go back to Agmon [1], but not sharp. • For eigenfunctions, the decay is governed by the uncertainty principle - see ADFJM ‘ Localization of eigenfunctions via an effective potential ’[2] • In [9], Shen proved that if V ∈ RH n 2 , then the fundamental solution Γ to the classical Schr¨ odinger operator − ∆ + V satisfies the bounds c 1 e − ε 1 d ( x,y,V ) ≤ Γ( x, y ) ≤ c 2 e − ε 2 d ( x,y,V ) , (2) | x − y | n − 2 | x − y | n − 2 where d is a certain distance function depending on V . 3 Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨ odinger operators
Exponential Decay of electric Schr¨ odinger operators • First results expressing upper estimates on the solutions in terms of a certain distance associated to V go back to Agmon [1], but not sharp. • For eigenfunctions, the decay is governed by the uncertainty principle - see ADFJM ‘ Localization of eigenfunctions via an effective potential ’[2] • In [9], Shen proved that if V ∈ RH n 2 , then the fundamental solution Γ to the classical Schr¨ odinger operator − ∆ + V satisfies the bounds c 1 e − ε 1 d ( x,y,V ) ≤ Γ( x, y ) ≤ c 2 e − ε 2 d ( x,y,V ) , (2) | x − y | n − 2 | x − y | n − 2 where d is a certain distance function depending on V . 3 Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨ odinger operators
Exponential Decay of electric Schr¨ odinger operators • First results expressing upper estimates on the solutions in terms of a certain distance associated to V go back to Agmon [1], but not sharp. • For eigenfunctions, the decay is governed by the uncertainty principle - see ADFJM ‘ Localization of eigenfunctions via an effective potential ’[2] • In [9], Shen proved that if V ∈ RH n 2 , then the fundamental solution Γ to the classical Schr¨ odinger operator − ∆ + V satisfies the bounds c 1 e − ε 1 d ( x,y,V ) ≤ Γ( x, y ) ≤ c 2 e − ε 2 d ( x,y,V ) , (2) | x − y | n − 2 | x − y | n − 2 where d is a certain distance function depending on V . 3 Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨ odinger operators
Exponential Decay of electric Schr¨ odinger operators • A natural question is whether the sharp exponential decay found by Shen for the fundamental solution to − ∆ + V can be extended to the non self-adjoint setting − div A ∇ + V . • Moreover, we also wondered whether we can obtain exponential decay results for the fundamental solution to the magnetic Schr¨ odinger operator . 4 Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨ odinger operators
Exponential Decay of electric Schr¨ odinger operators • A natural question is whether the sharp exponential decay found by Shen for the fundamental solution to − ∆ + V can be extended to the non self-adjoint setting − div A ∇ + V . • Moreover, we also wondered whether we can obtain exponential decay results for the fundamental solution to the magnetic Schr¨ odinger operator . 4 Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨ odinger operators
The generalized magnetic Schr¨ odinger operator We consider the operator formally given by L = − ( ∇ − i a ) T A ( ∇ − i a ) + V, (3) where a = ( a 1 , . . . , a n ) is a vector of real-valued L 2 loc ( R n ) functions, A and V as before. Denote D a = ∇ − i a , and the magnetic field by B , so that B = curl a . (4) 5 Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨ odinger operators
Properties of the magnetic Schr¨ odinger operator • The magnetic Schr¨ odinger operator exhibits a property called gauge invariance : quantitative assumptions should be put on B rather than a . • The diamagnetic inequality � � � � � ∇| u | ( x ) � ≤ � D a u ( x ) � . (5) � � � � • When A ≡ I so that L M := L = ( ∇ − i a ) 2 + V , the operator L M is dominated by the Schr¨ odinger operator L E := − ∆ + V in the following sense: for each ε > 0 , | ( L M + ε ) − 1 f | ≤ ( − ∆ + ε ) − 1 | f | , for each f ∈ H = L 2 ( R n ) . (6) The above is known as the Kato-Simon inequality . 6 Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨ odinger operators
Properties of the magnetic Schr¨ odinger operator • The magnetic Schr¨ odinger operator exhibits a property called gauge invariance : quantitative assumptions should be put on B rather than a . • The diamagnetic inequality � � � � � ∇| u | ( x ) � ≤ � D a u ( x ) � . (5) � � � � • When A ≡ I so that L M := L = ( ∇ − i a ) 2 + V , the operator L M is dominated by the Schr¨ odinger operator L E := − ∆ + V in the following sense: for each ε > 0 , | ( L M + ε ) − 1 f | ≤ ( − ∆ + ε ) − 1 | f | , for each f ∈ H = L 2 ( R n ) . (6) The above is known as the Kato-Simon inequality . 6 Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨ odinger operators
Properties of the magnetic Schr¨ odinger operator • The magnetic Schr¨ odinger operator exhibits a property called gauge invariance : quantitative assumptions should be put on B rather than a . • The diamagnetic inequality � � � � � ∇| u | ( x ) � ≤ � D a u ( x ) � . (5) � � � � • When A ≡ I so that L M := L = ( ∇ − i a ) 2 + V , the operator L M is dominated by the Schr¨ odinger operator L E := − ∆ + V in the following sense: for each ε > 0 , | ( L M + ε ) − 1 f | ≤ ( − ∆ + ε ) − 1 | f | , for each f ∈ H = L 2 ( R n ) . (6) The above is known as the Kato-Simon inequality . 6 Authors: Svitlana Mayboroda, Bruno Poggi Exponential decay for fundamental solutions of Schr¨ odinger operators
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