Introduction Optimalization of the ground state eigenvalue Optimization with respect to ω Domain comparison Differential Operators on Graphs and Waveguides Graz University of Technology February 26th, 2019 Spectral geometry in a rotating frame: properties of the ground state Diana Barseghyan Nuclear Physics Institute of the ASCR, ˇ Reˇ z near Prague & University of Ostrava joint work with Pavel Exner Diana Barseghyan 1/23
Introduction Optimalization of the ground state eigenvalue Optimization with respect to ω Domain comparison We consider the spectral properties of the operator (formally) defined by � � H ω ( x 0 , y 0 ) = − ∆ + i ω ( x − x 0 ) ∂ y − ( y − y 0 ) ∂ x on Ω ⊂ R 2 subject to the Dirichlet boundary conditions. Here ( x 0 , y 0 ) ∈ R 2 . ω > 0 , Physical motivations The above operator describes a quantum particle confined to a planar domain Ω rotating around a fixed point with an angular velocity ω . Quantum effects associated with rotation attracted a particular attention in connection with properties of ultracold gases. Diana Barseghyan 2/23
Introduction Optimalization of the ground state eigenvalue Optimization with respect to ω Domain comparison We consider the spectral properties of the operator (formally) defined by � � H ω ( x 0 , y 0 ) = − ∆ + i ω ( x − x 0 ) ∂ y − ( y − y 0 ) ∂ x on Ω ⊂ R 2 subject to the Dirichlet boundary conditions. Here ( x 0 , y 0 ) ∈ R 2 . ω > 0 , Physical motivations The above operator describes a quantum particle confined to a planar domain Ω rotating around a fixed point with an angular velocity ω . Quantum effects associated with rotation attracted a particular attention in connection with properties of ultracold gases. Diana Barseghyan 2/23
Introduction Optimalization of the ground state eigenvalue Optimization with respect to ω Domain comparison Associated quadratic form For any u ∈ C ∞ 0 (Ω) one has ( H ω ( x 0 , y 0 ) u , u ) L 2 (Ω) = � � � � d x d y − ω 2 2 � � (( x − x 0 ) 2 +( y − y 0 ) 2 ) | u | 2 d x d y , � i ∇ u + � Au � 4 Ω Ω where � A = ( − y + y 0 , x − x 0 ) . Boundedness of Ω implies that the corresponding operator is bounded from below, hence it allows for Friedrichs extension � � 2 � ( x − x 0 ) 2 + ( y − y 0 ) 2 � − ω 2 i ∇ + ω � � H ω ( x 0 , y 0 ) = A 2 4 with the domain H 2 (Ω) ∩ H 1 0 (Ω) . Diana Barseghyan 3/23
Introduction Optimalization of the ground state eigenvalue Optimization with respect to ω Domain comparison Associated quadratic form For any u ∈ C ∞ 0 (Ω) one has ( H ω ( x 0 , y 0 ) u , u ) L 2 (Ω) = � � � � d x d y − ω 2 2 � � (( x − x 0 ) 2 +( y − y 0 ) 2 ) | u | 2 d x d y , � i ∇ u + � Au � 4 Ω Ω where � A = ( − y + y 0 , x − x 0 ) . Boundedness of Ω implies that the corresponding operator is bounded from below, hence it allows for Friedrichs extension � � 2 � ( x − x 0 ) 2 + ( y − y 0 ) 2 � − ω 2 i ∇ + ω � � H ω ( x 0 , y 0 ) = A 2 4 with the domain H 2 (Ω) ∩ H 1 0 (Ω) . Diana Barseghyan 3/23
Introduction Optimalization of the ground state eigenvalue Optimization with respect to ω Domain comparison Remark By simple gauge transformation, namely u ( x , y ) �→ u ( x , y ) e − i ω ( xy 0 − yx 0 ) / 2 , the operator � H ω ( x 0 , y 0 ) is unitarily equivalent to � � 2 � ( x − x 0 ) 2 + ( y − y 0 ) 2 � − ω 2 i ∇ + ω � H ω ( x 0 , y 0 ) = 2 A 4 with A := ( − y , x ) . Diana Barseghyan 4/23
Introduction Optimalization of the ground state eigenvalue Optimization with respect to ω Domain comparison The spectrum of � H ω ( x 0 , y 0 ) is purely discrete. The main object of interest in the talk Our concern will be the principal eigenvalue λ ω 1 ( x 0 , y 0 ) of � H ω ( x 0 , y 0 ) . Diana Barseghyan 5/23
Introduction Existence and uniqueness of maximum, absence of minimums Optimalization of the ground state eigenvalue Convex sets Optimization with respect to ω Slow rotation Domain comparison The first problem concerns ( x 0 , y 0 ) �→ λ ω 1 ( x 0 , y 0 ) for fixed Ω and ω , in particular, the existence of its extrema. Theorem (B.-Exner, 2019) 1 ( · , · ) as a map R 2 → R has no minima. It has a unique λ ω maximum. Remark λ ω 1 ( x 0 , y 0 ) → −∞ holds as ( x 0 , y 0 ) → ∞ . This guarantees the existence of maxima. Diana Barseghyan 6/23
Introduction Existence and uniqueness of maximum, absence of minimums Optimalization of the ground state eigenvalue Convex sets Optimization with respect to ω Slow rotation Domain comparison The first problem concerns ( x 0 , y 0 ) �→ λ ω 1 ( x 0 , y 0 ) for fixed Ω and ω , in particular, the existence of its extrema. Theorem (B.-Exner, 2019) 1 ( · , · ) as a map R 2 → R has no minima. It has a unique λ ω maximum. Remark λ ω 1 ( x 0 , y 0 ) → −∞ holds as ( x 0 , y 0 ) → ∞ . This guarantees the existence of maxima. Diana Barseghyan 6/23
Introduction Existence and uniqueness of maximum, absence of minimums Optimalization of the ground state eigenvalue Convex sets Optimization with respect to ω Slow rotation Domain comparison Sketch of the proof Let ( x 0 , y 0 ) be a possible extrema point. Step 1 We employ normalized eigenfunctions u ( x 0 , y 0 ) and v ( x 0 , y 0 ) ω ω corresponding to λ ω 1 ( x 0 , y 0 ) such that u ( x 0 + t , y 0 ) = u ( x 0 , y 0 ) + O ( t ) , ω ω v ( x 0 , y 0 + s ) = v ( x 0 , y 0 ) + O ( s ) , ω ω for small values of t and s , where the error term is understood in the L ∞ sense. The existence of such eigenfunctions is due to [N. Raymond: Bound States of the Magnetic Schr¨ odinger Operators, EMS, 2017] 1 ( x 0 , y 0 ) is simple then u ( x 0 , y 0 ) = v ( x 0 , y 0 ) If the eigenvalue λ ω . In fact we ω ω shall see that this not true in general. Diana Barseghyan 7/23
Introduction Existence and uniqueness of maximum, absence of minimums Optimalization of the ground state eigenvalue Convex sets Optimization with respect to ω Slow rotation Domain comparison Step 2 The key point – to prove the following implication: ( x 0 , y 0 ) is an extremum point ⇓ � �� � � � | 2 d x d y = 0 | 2 d x d y = 0 ( x − x 0 ) | u ( x 0 , y 0 ) ( y − y 0 ) | v ( x 0 , y 0 ) and ω ω Ω Ω Idea of its proof: assume that this in not true, for example, one has � | 2 d x d y > 0 . ( x − x 0 ) | u ( x 0 , y 0 ) ω Ω Using min-max principle, one can show that the above inequality implies for any h < 0 small enough λ ω 1 ( x 0 + h , y 0 ) < λ ω 1 ( x 0 , y 0 ) . Also, using min-max principle, one can deduce for small t > 0 λ ω 1 ( x 0 , y 0 ) < λ ω 1 ( x 0 + t , y 0 ) . Contradiction (since ( x 0 , y 0 ) is an extremum). Diana Barseghyan 8/23
Introduction Existence and uniqueness of maximum, absence of minimums Optimalization of the ground state eigenvalue Convex sets Optimization with respect to ω Slow rotation Domain comparison Step 2 The key point – to prove the following implication: ( x 0 , y 0 ) is an extremum point ⇓ � �� � � � | 2 d x d y = 0 | 2 d x d y = 0 ( x − x 0 ) | u ( x 0 , y 0 ) ( y − y 0 ) | v ( x 0 , y 0 ) and ω ω Ω Ω Idea of its proof: assume that this in not true, for example, one has � | 2 d x d y > 0 . ( x − x 0 ) | u ( x 0 , y 0 ) ω Ω Using min-max principle, one can show that the above inequality implies for any h < 0 small enough λ ω 1 ( x 0 + h , y 0 ) < λ ω 1 ( x 0 , y 0 ) . Also, using min-max principle, one can deduce for small t > 0 λ ω 1 ( x 0 , y 0 ) < λ ω 1 ( x 0 + t , y 0 ) . Contradiction (since ( x 0 , y 0 ) is an extremum). Diana Barseghyan 8/23
Introduction Existence and uniqueness of maximum, absence of minimums Optimalization of the ground state eigenvalue Convex sets Optimization with respect to ω Slow rotation Domain comparison Step 3 Using (cf. Step 2) � � | 2 d x d y = 0 , | 2 d x d y = 0 ( x − x 0 ) | u ( x 0 , y 0 ) ( y − y 0 ) | v ( x 0 , y 0 ) ω ω Ω Ω and min-max principle one can prove that for all nonzero and sufficiently small h λ ω 1 ( x 0 + h , y 0 ) < λ ω 1 ( x 0 , y 0 ) . Thus ( x 0 , y 0 ) is a point of maximum. Diana Barseghyan 9/23
Introduction Existence and uniqueness of maximum, absence of minimums Optimalization of the ground state eigenvalue Convex sets Optimization with respect to ω Slow rotation Domain comparison Theorem (B.-Exner, 2019) Let Ω be convex, then ( x 0 , y 0 ) �→ λ ω 1 ( x 0 , y 0 ) reaches its maximum at a point belonging to Ω . Diana Barseghyan 10/23
Introduction Existence and uniqueness of maximum, absence of minimums Optimalization of the ground state eigenvalue Convex sets Optimization with respect to ω Slow rotation Domain comparison If ω is small then the position of the maximum can be described more precisely. Definition Given a region Σ ⊂ R 2 and a line P , we denote by Σ P the mirror image of Σ with respect to P . Theorem (B.-Exner, 2019) Let Ω be convex set and P be a line which divides Ω into two parts, Ω 1 and Ω 2 , in such a way that Ω P 1 ⊂ Ω 2 . Then for small enough values of ω the point at which λ ω 1 ( x 0 , y 0 ) attains its maximum does not belong to Ω 1 . Diana Barseghyan 11/23
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