On continuous spectrum of magnetic Schrödinger operators on periodic discrete graphs
aa r X i v : . [ m a t h . SP ] J a n ON CONTINUOUS SPECTRUM OF MAGNETIC SCHR ¨ODINGEROPERATORS ON PERIODIC DISCRETE GRAPHS
EVGENY KOROTYAEV AND NATALIA SABUROVA
Abstract.
We consider Schr¨odinger operators with periodic electric and magnetic potentialson periodic discrete graphs. The spectrum of such operators consists of an absolutely con-tinuous (a.c.) part (a union of a finite number of non-degenerate bands) and a finite numberof eigenvalues of infinite multiplicity. We prove the following results: 1) the a.c. spectrumof the magnetic Schr¨odinger operators is empty for specific graphs and magnetic fields; 2)we obtain necessary and sufficient conditions under which the a.c. spectrum of the magneticSchr¨odinger operators is empty; 3) the spectrum of the magnetic Schr¨odinger operator witheach magnetic potential tα , where t is a coupling constant, has an a.c. component for allexcept finitely many t from any bounded interval. Introduction and main results
There are many physical phenomena described by periodic Schr¨odinger operators. In gen-eral, the spectral analysis of a periodic operator H is based on the so-called Floquet decom-position, where this operator has a representation as a direct integral of a family of fiberoperators H ( k ) with discrete spectra, see e.g., [12]. The fiber operator H ( k ) depends on theso-called quasimomentum k belonging to the torus T d = R d / (2 π Z ) d . Its eigenvalues λ j ( k ), j = 1 , , . . . , arranged in non-decreasing order, depend on k continuously. The spectrum ofthe operator H is a union of spectral bands σ j , arising as the ranges of the band functions λ j ( · ), i.e., σ j = λ j ( T d ). If some function λ j ( k ) = Λ j = const on an open domain of T d , thenΛ j is an eigenvalue of H of infinite multiplicity. We call { Λ j } a flat band or a degenerate band.All other bands are non-degenerate . The union of all flat bands is the flat spectrum of H andthe union of all non-degenerate bands is the a.c. spectrum of H . One of important problemsis to describe the spectrum: when the operator H has the flat spectrum and when it doesnot. There are a lot of results about the spectrum of periodic differential operators, see, e.g.,[1, 2, 13, 14].There are some results about the spectrum of discrete Schr¨odinger operators with periodicelectric potentials on periodic graphs, see [6, 9, 11]. It is shown in [11] that the first band ofthese operators is non-degenerate. Thus, the spectrum of the Schr¨odinger operators alwayshas an a.c. component. Some classes of periodic graphs on which the discrete Laplacian hasonly the a.c. spectrum are described in [6].We consider discrete Schr¨odinger operators with periodic magnetic and electric potentialson periodic graphs. This case and even the case of magnetic Laplacians are much morecomplicated compared to the non-magnetic one. We do not know results about the flatspectrum of the discrete magnetic Schr¨odinger operators. Our goal is to show that, in contrast Date : January 15, 2021.
Key words and phrases. discrete magnetic Schr¨odinger operators, periodic graphs, periodic electric andmagnetic potentials, absolutely continuous spectrum, flat bands. to the non-magnetic case, the a.c. spectrum of the magnetic Schr¨odinger operators may beempty (for specific graphs and magnetic potentials) and to determine necessary and sufficientconditions for absence of the a.c. spectrum. Moreover, we show that absence of the a.c.spectrum is a quite rare situation, i.e., the spectrum of the magnetic Schr¨odinger operator witha periodic magnetic potential tα , where t ∈ R is a coupling constant, has an a.c. componentfor all except finitely many t from any bounded interval.1.1. Magnetic Schr¨odinger operators on periodic graphs.
Let G = ( V , E ) be a con-nected infinite graph, possibly having loops and multiple edges and embedded into the space R d . Here V is the set of its vertices and E is the set of its unoriented edges. Considering eachedge in E to have two orientations, we introduce the set A of all oriented edges. An edgestarting at a vertex u and ending at a vertex v from V will be denoted as the ordered pair( u, v ) ∈ A and is said to be incident to the vertices. Let e = ( v, u ) be the inverse edge of e = ( u, v ) ∈ A . We define the degree κ v of the vertex v ∈ V as the number of all edges from A , starting at v .Let Γ be a lattice of rank d in R d with a basis { a , . . . , a d } , i.e.,Γ = n a : a = d X s =1 n s a s , n s ∈ Z , s ∈ N d o , N d = { , . . . , d } , and let Ω = n x ∈ R d : x = d X s =1 t s a s , t s < , s ∈ N d o (1.1)be the fundamental cell of the lattice Γ. We define the equivalence relation on R d : x ≡ y (mod Γ) ⇔ x − y ∈ Γ ∀ x, y ∈ R d . We consider locally finite Γ -periodic graphs G , i.e., graphs satisfying the following conditions: • G = G + a for any a ∈ Γ and the quotient graph G ∗ = G / Γ is finite.The basis a , . . . , a d of the lattice Γ is called the periods of G . We call the quotient graph G ∗ = G / Γ the fundamental graph of the periodic graph G . The fundamental graph G ∗ is agraph on the d -dimensional torus R d / Γ. The graph G ∗ = ( V ∗ , E ∗ ) has the vertex set V ∗ = V / Γ,the set E ∗ = E / Γ of unoriented edges and the set A ∗ = A / Γ of oriented edges which are finite.Let ℓ ( V ) be the Hilbert space of all square summable functions f : V → C equipped withthe norm k f k ℓ ( V ) = X v ∈V | f ( v ) | < ∞ . We consider the magnetic Schr¨odinger operators H α acting on ℓ ( V ) and given by H α = ∆ α + Q, (1.2)where Q is an electric potential and ∆ α is the magnetic Laplacian having the form(∆ α f )( v ) = X e =( v, u ) ∈A (cid:0) f ( v ) − e iα ( e ) f ( u ) (cid:1) , f ∈ ℓ ( V ) , v ∈ V , (1.3) α : A → R is a magnetic vector potential on G , i.e., it satisfies α ( e ) = − α ( e ) , ∀ e ∈ A . (1.4) N SPECTRUM OF MAGNETIC SCHR ¨ODINGER OPERATORS ON PERIODIC GRAPHS 3
The sum in (1.3) is taken over all edges from A starting at the vertex v . We assume that themagnetic potential α and the electric potential Q are Γ-periodic, i.e., they satisfy α ( e + a ) = α ( e ) , Q ( v + a ) = Q ( v ) , ∀ ( v, e , a ) ∈ V × A × Γ . (1.5)It is known that the magnetic Schr¨odinger operator H α is a bounded self-adjoint operator on ℓ ( V ) (see, e.g., [5]).1.2. Spectrum of the magnetic Schr¨odinger operator.
The magnetic Schr¨odinger op-erator H α on ℓ ( V ) has the decomposition into a constant fiber direct integral given by H = 1(2 π ) d Z ⊕ T d ℓ ( V ∗ ) dk, U H α U − = 1(2 π ) d Z ⊕ T d H α ( k ) dk, (1.6) T d = R d / (2 π Z ) d , for some unitary operator U : ℓ ( V ) → H . Here ℓ ( V ∗ ) = C ν is the fiberspace, ν = V ∗ . The parameter k ∈ T d is called the quasimomentum . The precise expressionof the fiber operator H α ( k ) is given by (2.15) – (2.16). Note that H α (0) is the magneticSchr¨odinger operator on G ∗ . Each fiber operator H α ( k ), k ∈ T d , has ν real eigenvalues λ α,j ( k ), j ∈ N ν , which are labeled in non-decreasing order (counting multiplicities) by λ α, ( k ) λ α, ( k ) . . . λ α,ν ( k ) , ∀ k ∈ T d . (1.7)Each λ α,j ( · ), j ∈ N ν , is a real and piecewise analytic function on the torus T d and creates the spectral band σ j ( H α ) given by σ j ( H α ) = [ λ − α,j , λ + α,j ] = λ α,j ( T d ) . (1.8)Note that if λ α,j ( · ) = Λ j = const on some subset of T d of positive Lebesgue measure, thenthe operator H α on G has the eigenvalue Λ j of infinite multiplicity. We call { Λ j } a flat band .Thus, the spectrum of the operator H α on the periodic graph G is given by σ ( H α ) = [ k ∈ T d σ (cid:0) H α ( k ) (cid:1) = ν [ j =1 σ j ( H α ) = σ ac ( H α ) ∪ σ fb ( H α ) , (1.9)where σ ac ( H α ) is the a.c. spectrum, which is a union of non-degenerate bands, and σ fb ( H α )is the flat spectrum which is the set of all flat bands (eigenvalues of infinite multiplicity).It is known (see, e.g., Proposition 4.2 in [6]) that λ ∗ is an eigenvalue of infinite multiplicityof the operator H α if and only if λ ∗ is an eigenvalue of the operator H α ( k ) for all k ∈ T d . Itgives another labeling of the eigenvalues of H α ( k ). If the operator H α has r > µ µ . . . µ r , r ν, (1.10)then we can separate constant eigenvalues µ j ( · ) = µ j = const, of H α ( · ), j ∈ N r . All othereigenvalues µ j ( k ), r < j ν , of H α ( k ) can be enumerated in non-decreasing order by µ r +1 ( k ) µ r +2 ( k ) . . . µ ν ( k ) , ∀ k ∈ T d , (1.11)counting multiplicities. We define the spectral bands S j ( H α ) = [ µ − j , µ + j ] = µ j ( T d ), j ∈ N ν ,where each band S j ( H α ), r < j ν , is non-degenerate, i.e., µ − j < µ + j . Thus, the number ofnon-degenerate spectral bands of the operator H α is ν − r . EVGENY KOROTYAEV AND NATALIA SABUROVA
Main results.
Recall that the a.c. spectrum of the Schr¨odinger operator H with themagnetic potential α = 0 is not empty, see [11]. The following simple examples show thatthis is not true for the magnetic Schr¨odinger operator H α , i.e., the a.c. spectrum of H α on G is empty for specific periodic graphs and magnetic potentials α . In the first example weconstruct the magnetic Schr¨odinger operators H α when their fiber operators H α ( k ) do notdepend on the quasimomentum k ∈ T d . It is clear that in this case the spectrum of H α is flat,since all band functions are constants. Example 1.1.
Let each oriented edge of a periodic graph G have multiplicity 2, and let aperiodic magnetic vector potential α on G satisfy | α ( e ) − α ( e e ) | = π for each pair of multiple edges ( e , e e ) ∈ A . (1.12) Then for any periodic electric potential Q the magnetic Schr¨odinger operator H α = ∆ α + Q hasthe decomposition (1.6), where its fiber operator has the form H α ( k ) = diag (cid:0) κ v + Q ( v ) (cid:1) v ∈V ∗ ,and κ v is the degree of the vertex v ∈ V ∗ . In particular, the spectrum of H α is flat and isgiven by (cid:8) κ v + Q ( v ) (cid:9) v ∈V ∗ . The second example shows that the condition H α ( · ) = const is not necessary for absence ofthe a.c. spectrum of H α . r r e − , e − , e − , e , e , e , e , e , r r r r − − G a ) G ∗ b ) r r v v ✲✲✲ e e e Figure 1. a ) The periodic graph G = ( Z , E ); b ) the fundamental graph G ∗ = G / (2 Z ). Example 1.2.
We consider the periodic graph G = ( Z , E ) , where the edge set is given by E = (cid:8) e n, = ( n, n + 1) for all n ∈ Z (cid:9) ∪ (cid:8) e n, = ( n, n + 1) for all even n ∈ Z (cid:9) (see Fig.1a). Let H α = ∆ α + Q be the Schr¨odinger operator on G , where a 2-periodic electricpotential Q and a periodic magnetic potential α are given by ( Q (0) = 0 Q (1) ∈ R , α ( e n,s ) = α o , if s = 1 and n is even π + α o , if s = 2 and n is even , if s = 1 and n is odd , n ∈ Z , s = 1 , , (1.13) and α o ∈ R . Then the fiber operator H α ( k ) depends on k ∈ T d , but the spectrum of H α is flatand is given by λ s = 3 + (cid:0) Q (1) + ( − s p Q (1) + 4 (cid:1) , s = 1 , . (1.14)The proof of Examples 1.1 and 1.2 is given in Subsection 2.4. Remark.
Below in Theorem 2.5 we determine necessary and sufficient conditions when thespectrum of the magnetic Schr¨odinger operator H α is flat, i.e., the a.c. spectrum of H α isempty. In order to present this result we need more definitions. N SPECTRUM OF MAGNETIC SCHR ¨ODINGER OPERATORS ON PERIODIC GRAPHS 5
Examples 1.1 and 1.2 show that all spectral bands of the magnetic Schr¨odinger operator H α may be flat. In the following theorem we prove that this is a quite rare situation. Theorem 1.3.
Let H tα = ∆ tα + Q be the magnetic Schr¨odinger operator defined by (1.2) –(1.3) with a periodic magnetic potential tα and a periodic electric potential Q on a periodicgraph G , where t is a coupling constant. Then the a.c. spectrum of H tα is not empty for allexcept finitely many t ∈ [0 , . Remarks.
1) It is known [1, 13] that the spectrum of the Schr¨odinger operator with periodicelectric and magnetic potentials on L ( R d ) is a.c. and the proof is based on the Thomas’approach. We do not use this argument. In order to prove Theorem 1.3 we determine traceformulas for the magnetic Schr¨odinger operator (see Theorem 2.4) as Fourier series and useclassical function theory. Moreover, as an unperturbed case we use the result from [11] thatthe first spectral band of the Schr¨odinger operator without magnetic field is not flat.2) Theorem 1.3 also holds true for weighted magnetic Laplace and Schr¨odinger operators,in particular, for the normalized ones. The proof repeats the proof of Theorem 1.3.1.4. Historical review.
There are a lot papers about the spectrum of periodic differentialoperators, see articles [1, 2, 4, 13, 14] and the references therein. The first result in thisdirection was obtained by Thomas [14]. He proved that each band function of Schr¨odingeroperators with periodic potentials on L ( R d ) is not flat. Later on this approach was used inmany papers [1, 2, 4, 13]. Danilov [2] proved that the spectrum of the Dirac operator withperiodic potentials on R d is a.c. Hempel and Herbst [4] proved that the spectrum of magneticLaplacians with small periodic magnetic vector potentials in R d is a.c. Birman and Suslina [1](for the case d = 2), and Sobolev [13] (for d >
2) proved that the spectrum of the Schr¨odingeroperator with periodic electric and magnetic potentials on L ( R d ) is a.c.In the discrete settings the situation is quite different. The spectrum of the discreteSchr¨odinger operator with periodic electric and magnetic potentials on periodic graphs con-sists of a finite number of bands. Some of them may be degenerate. Thus, the spectrum of thediscrete Schr¨odinger operators has an a.c. component which is a union of all non-degeneratebands and a flat component which is the set of all degenerate bands (eigenvalues of infinitemultiplicity). In [11] it was proved that the first band of the discrete Schr¨odinger opera-tors with periodic electric potentials on periodic graphs is always non-degenerate, i.e., thea.c. spectrum is not empty. It was also shown that all except two bands of the Schr¨odingeroperators may be degenerate (see Proposition 7.2 in [9]) and the number of the flat bandsmay be arbitrary. We do not know results about the flat spectrum of the discrete magneticSchr¨odinger operators. 2. Proof
Edge indices.
For each x ∈ R d we introduce the vector x A ∈ R d by x A = ( x , . . . , x d ) , where x = d P s =1 x s a s , (2.1)i.e., x A is the coordinate vector of x with respect to the basis A = { a , . . . , a d } of the latticeΓ.For any vertex v ∈ V of a Γ-periodic graph G the following unique representation holds true: v = v + [ v ] , where v ∈ Ω , [ v ] ∈ Γ , (2.2) EVGENY KOROTYAEV AND NATALIA SABUROVA
Ω is a fundamental cell of the lattice Γ defined by (1.1). In other words, each vertex v canbe obtained from a vertex v ∈ Ω by a shift by a vector [ v ] ∈ Γ. For any oriented edge e = ( u, v ) ∈ A we define the edge index τ ( e ) as the vector of the lattice Z d given by τ ( e ) = [ v ] A − [ u ] A ∈ Z d , (2.3)where [ v ] ∈ Γ is defined by (2.2) and the vector [ v ] A ∈ Z d is given by (2.1).The edge indices τ ( e ) and the magnetic potential α are Γ-periodic, i.e., they satisfy τ ( e + a ) = τ ( e ) , α ( e + a ) = α ( e ) , ∀ ( e , a ) ∈ A × Γ . (2.4)On the set A of all oriented edges of the Γ-periodic graph G we define the surjection f : A → A ∗ = A / Γ , (2.5)which maps each e ∈ A to its equivalence class e ∗ = f ( e ) which is an oriented edge of thefundamental graph G ∗ . The identities (2.4) and the mapping f allow us to define uniquely • the magnetic potential α on edges of the fundamental graph G ∗ = ( V ∗ , A ∗ ) which is inducedby the magnetic potential α : α ( e ∗ ) = α ( e ) for some e ∈ A such that e ∗ = f ( e ) , e ∗ ∈ A ∗ ; (2.6) • indices of the fundamental graph edges which are induced by the indices of the periodicgraph edges: τ ( e ∗ ) = τ ( e ) for some e ∈ A such that e ∗ = f ( e ) , e ∗ ∈ A ∗ . (2.7)2.2. Cycle indices and magnetic fluxes. A path p in a graph G = ( V , A ) is a sequence ofconsecutive edges p = ( e , e , . . . , e n ) , (2.8)where e s = ( v s − , v s ) ∈ A , s = 1 , . . . , n , for some vertices v , v , . . . , v n ∈ V . The vertices v and v n are called the initial and terminal vertices of the path p , respectively. If v = v n , thenthe path p is called a cycle . The number n of edges in a cycle c is called the length of c andis denoted by | c | , i.e., | c | = n . Remark.
A path p is uniquely defined by the sequence of it’s oriented edges ( e , e , . . . , e n ).The sequence of it’s vertices ( v , v , . . . , v n ) does not uniquely define p , since multiple edgesare allowed in the graph G .Let C be the sets of all cycles of the fundamental graph G ∗ . For any cycle c ∈ C we define • the cycle index τ ( c ) ∈ Z d by τ ( c ) = X e ∈ c τ ( e ) , c ∈ C , (2.9) • the flux α ( c ) ∈ [ − π, π ] of the magnetic potential α through the cycle c by α ( c ) = (cid:18) X e ∈ c α ( e ) (cid:19) mod 2 π, c ∈ C . (2.10)Note that we consider fluxes of the magnetic potential α modulo 2 π , since the magneticpotential α appears in the Laplacian ∆ α via the factor e iα ( e ) , e ∈ A . For each cycle c = N SPECTRUM OF MAGNETIC SCHR ¨ODINGER OPERATORS ON PERIODIC GRAPHS 7 ( e , . . . , e n ) we define a cycle c = ( e n , . . . , e ). From the definition of indices and fluxes itfollows that τ ( e ) = − τ ( e ) , α ( e ) = − α ( e ) , ∀ e ∈ A ∗ ; τ ( c ) = − τ ( c ) , α ( c ) = − α ( c ) , ∀ c ∈ C . (2.11)2.3. Direct integral.
We introduce the Hilbert space, i.e., a constant fiber direct integral, H = L (cid:16) T d , dk (2 π ) d , ℓ ( V ∗ ) (cid:17) = Z ⊕ T d ℓ ( V ∗ ) dk (2 π ) d , T d = R d / (2 π Z ) d , (2.12)equipped with the norm k g k H = R T d k g ( k, · ) k ℓ ( V ∗ ) dk (2 π ) d , where the function g ( k, · ) ∈ ℓ ( V ∗ )for almost all k ∈ T d . We identify the vertices of the fundamental graph G ∗ = ( V ∗ , E ∗ ) withthe vertices from the fundamental cell Ω. The unitary Gelfand transform U : ℓ ( V ) → H isgiven by( U f )( k, v ) = X m=( m ,...,m d ) ∈ Z d e − i h m ,k i f ( v + m a + . . . + m d a d ) , ( k, v ) ∈ T d × V ∗ , (2.13)where { a , . . . , a d } is the basis of the lattice Γ, and h· , · i denotes the standard inner productin R d . We recall Theorem 3.1 from [10]. Theorem 2.1.
The magnetic Schr¨odinger operator H α = ∆ α + Q on ℓ ( V ) has the followingdecomposition into a constant fiber direct integral U H α U − = Z ⊕ T d H α ( k ) dk (2 π ) d , (2.14) where U : ℓ ( V ) → H is the unitary Gelfand transform defined by (2.13), and the fibermagnetic Schr¨odinger operator H α ( k ) on ℓ ( V ∗ ) has the form H α ( k ) = ∆ α ( k ) + Q, ∀ k ∈ T d . (2.15) Here Q is the electric potential on ℓ ( V ∗ ) and ∆ α ( k ) is the fiber magnetic Laplacian given by (cid:0) ∆ α ( k ) f (cid:1) ( v ) = X e =( v, u ) ∈A ∗ (cid:0) f ( v ) − e i ( α ( e )+ h τ ( e ) , k i ) f ( u ) (cid:1) , f ∈ ℓ ( V ∗ ) , v ∈ V ∗ , (2.16) where τ ( e ) is the index of the edge e ∈ A ∗ defined by (2.3), (2.7). From Theorem 2.1 it follows that the fiber magnetic Schr¨odinger operator H α ( k ) is the ν × ν matrix defined by (2.15) – (2.16) and in the standard orthonormal basis of ℓ ( V ∗ ) = C ν , ν = V ∗ , is given by H α ( k ) = q − A α ( k ) , q = diag( q v ) v ∈V ∗ , q v = κ v + Q ( v ) , (2.17)where κ v is the degree of the vertex v , and the matrix A α ( k ) = (cid:0) A α,uv ( k ) (cid:1) u,v ∈V ∗ has the form A α,uv ( k ) = X e =( u,v ) ∈A ∗ e − i ( α ( e )+ h τ ( e ) , k i ) . (2.18) Lemma 2.2.
Let H α ( k ) be the fiber operator given by (2.17), (2.18). Then λ ∗ is an eigen-value of infinite multiplicity of the magnetic Schr¨odinger operator H α if and only if λ ∗ is aneigenvalue of H α ( k ) for all k ∈ T d . EVGENY KOROTYAEV AND NATALIA SABUROVA
Proof.
The proof is quite standard. But for the reader’s convenience we repeat it. Let λ ∗ be an eigenvalue of infinite multiplicity of H α . Then det (cid:0) λ ∗ I ν − H α ( k ) (cid:1) = 0 for all k ∈ B ,where B is a subset of T d of positive Lebesgue measure, and I ν is the identity ν × ν matrix.The function f ( k ) = det (cid:0) λ ∗ I ν − H α ( k ) (cid:1) is real analytic in k ∈ T d (moreover, it is an entirefunction of k ∈ C d ). Then f ( k ) = 0 for all k ∈ T d , i.e., λ ∗ is an eigenvalue of H α ( k ) for all k ∈ T d . The converse is obvious.2.4. Proof of Examples.
We prove Examples 1.1 and 1.2 where we construct magneticSchr¨odinger operators with the empty a.c. spectrum.
Proof of Example 1.1.
Let each oriented edge of a periodic graph G have multiplicity 2,and let a magnetic vector potential α on G satisfy (1.12). Then for each edge e = ( u, v ) ∈ A ∗ of the fundamental graph G ∗ = ( V ∗ , A ∗ ) there exists an edge e e = ( u, v ) ∈ A ∗ such that τ ( e e ) = τ ( e ) , and | α ( e ) − α ( e e ) | = π. Thus, the matrix A α ( k ) = (cid:0) A α,uv ( k ) (cid:1) u,v ∈V ∗ given by (2.18) has the form A α,uv ( k ) = P e =( u,v ) ∈A ∗ e − i ( α ( e )+ h τ ( e ) , k i ) = P e =( u,v ) ∈A ∗ (cid:0) e − i ( α ( e )+ h τ ( e ) , k i ) + e − i ( α ( e e )+ h τ ( e e ) , k i ) (cid:1) = P e =( u,v ) ∈A ∗ e − i h τ ( e ) , k i (cid:0) e − iα ( e ) + e − i ( α ( e ) ± π ) (cid:1) = 0 . This and (2.17) yield that the fiber magnetic Schr¨odinger operator H α ( k ) is diagonal and hasthe form H α ( k ) = diag (cid:0) κ v + Q ( v ) (cid:1) v ∈V ∗ . Thus, the spectrum of H α on G is flat and is givenby (cid:8) κ v + Q ( v ) (cid:9) v ∈V ∗ . Proof of Example 1.2.
The fundamental graph G ∗ = G / (2 Z ) consists of two vertices v , v and three multiple edges e , e , e connecting these vertices (Fig. 1 b ) with indices τ ( e ) = τ ( e ) = 0 , τ ( e ) = 1 , and their inverse edges. The magnetic potential α on edges of G ∗ is given by α ( e ) = α o , α ( e ) = π + α o , α ( e ) = 0 , for some α o ∈ R . Then the fiber magnetic Schr¨odinger operator H α ( k ), k ∈ T , given by (2.17)– (2.18), on G ∗ has the form H α ( k ) = (cid:18) − e − iα − e − i ( π + α o ) − e − ik − e iα − e i ( π + α ) − e ik Q (1) (cid:19) = (cid:18) − e − ik − e ik Q (1) (cid:19) , where Q (0) = 0. Thus, the eigenvalues of H α ( k ) are given by (1.14).2.5. Trace formulas.
In order to formulate trace formulas for the fiber magnetic Schr¨odingeroperator H α ( k ), we need some modifications of the fundamental graph G ∗ . We add a loop e v with index τ ( e v ) = 0 and the magnetic potential α ( e v ) = 0 at each vertex v of the fundamentalgraph G ∗ = ( V ∗ , A ∗ ) and consider the modified fundamental graph e G ∗ = ( V ∗ , e A ∗ ), where e A ∗ = A ∗ ∪ { e v } v ∈V ∗ . (2.19)We denote by e C the set of all cycles on e G ∗ . For each cycle c ∈ e C we define the weight ω ( c ) = ω ( e ) . . . ω ( e n ) , where c = ( e , . . . , e n ) ∈ e C , (2.20) N SPECTRUM OF MAGNETIC SCHR ¨ODINGER OPERATORS ON PERIODIC GRAPHS 9 and ω ( e ) is defined by ω ( e ) = (cid:26) − , if e ∈ A ∗ q v , if e = e v , q v = κ v + Q ( v ) . (2.21)The next lemma shows that the operator H α ( k ) can be considered as a fiber weightedmagnetic operator on the modified fundamental graph e G ∗ = ( V ∗ , e A ∗ ). Lemma 2.3.
The fiber magnetic Schr¨odinger operator H α ( k ) = (cid:0) H α,uv ( k ) (cid:1) u,v ∈V ∗ given by(2.17), (2.18) satisfies H α,uv ( k ) = X e =( u,v ) ∈ e A ∗ ω ( e ) e − i ( α ( e )+ h τ ( e ) , k i ) , ∀ u, v ∈ V ∗ , ∀ k ∈ T d , (2.22) where e A ∗ is the set of all edges of the modified fundamental graph e G ∗ defined by (2.19); ω ( e ) is given by (2.21), and τ ( e ) is the index of the edge e ∈ A ∗ defined by (2.3), (2.7). Proof.
Let u, v ∈ V ∗ . If u = v , then, using (2.21) and (2.17), (2.18), we have X e =( u,v ) ∈ e A ∗ ω ( e ) e − i ( α ( e )+ h τ ( e ) , k i ) = − X e =( u,v ) ∈A ∗ e − i ( α ( e )+ h τ ( e ) , k i ) = H α,uv ( k ) . Similarly, if u = v , then we obtain X e =( v,v ) ∈ e A ∗ ω ( e ) e − i ( α ( e )+ h τ ( e ) , k i ) = ω ( e v ) − X e =( v,v ) ∈A ∗ e − i ( α ( e )+ h τ ( e ) , k i ) = q v − X e =( v,v ) ∈A ∗ e − i ( α ( e )+ h τ ( e ) , k i ) = H α,vv ( k ) . Thus, the identity (2.22) has been proved.Let e C n, m be the set of all cycles from e C of length n and with index m ∈ Z d : e C n, m = { c ∈ e C : | c | = n and τ ( c ) = m } , (2.23)where | c | is the length of the cycle c , and τ ( c ) is the index of c defined by (2.9).In the following theorem we determine all Fourier coefficients of Tr H nα ( k ) as functions of k ∈ T d . This is a crucial point for our consideration. Theorem 2.4.
Let H α ( k ) , k ∈ T d , be the fiber magnetic Schr¨odinger operator defined by(2.15) – (2.16). Then for each n ∈ N the trace of H nα ( k ) has the following Fourier seriesrepresentation Tr H nα ( k ) = X m ∈ Z d h α,n, m e − i h m ,k i , h α,n, m = X c ∈ e C n, m ω ( c ) e − iα ( c ) , supp h α,n, · ⊂ { m ∈ Z d : k m k nτ + } , (2.24) where τ + = max e ∈A ∗ k τ ( e ) k , τ ( e ) is the index of the edge e ∈ A ∗ , and k · k is the standardnorm in R d . Here e C n, m is defined by (2.23); α ( c ) is the flux of the magnetic potential α throughthe cycle c defined by (2.9), and ω ( c ) is given by (2.20). Remark.
1) The formulas (2.24) are trace formulas , where the traces of the fiber operatorsare expressed in terms of some geometric parameters of the graph (vertex degrees, cycle indicesand lengths).2) There are trace formulas for Schr¨odinger operators with periodic potentials on the line,see e.g., [7]. They were used to obtain two-sided estimates of potentials in terms of gap lengths(or a solution for KdV in terms of the action variables) in [8] via the conformal mapping theoryfor the quasimomentum. Unfortunately, we do not know results about trace formulas for themultidimensional case.
Proof.
Using (2.22), for each n ∈ N we obtainTr H nα ( k ) = X v ,...,v n ∈V ∗ H α,v v ( k ) H α,v v ( k ) . . . H α,v n − v n ( k ) H α,v n v ( k )= X v ,...,v n ∈V ∗ X e ,... e n ∈ e A ∗ ω ( e ) ω ( e ) . . . ω ( e n ) e − i ( α ( e )+ α ( e )+ ... + α ( e n )+ h τ ( e )+ τ ( e )+ ... + τ ( e n ) ,k i ) = X c ∈ e C n ω ( c ) e − i ( α ( c )+ h τ ( c ) ,k i ) , where e j = ( v j , v j +1 ) , j ∈ N n , v n +1 = v , and e C n is the set of all cycles of length n on e G ∗ . Thus, we have the finite Fourier seriesfor 2 π Z d -periodic function Tr H nα ( k ), since τ ( c ) ∈ Z d . We rewrite this Fourier series in thestandard formTr H nα ( k ) = X c ∈ e C n ω ( c ) e − i ( α ( c )+ h τ ( c ) ,k i ) = X m ∈ Z d X c ∈ e C n, m ω ( c ) e − iα ( c ) e − i h m ,k i = X m ∈ Z d e − i h m ,k i X c ∈ e C n, m ω ( c ) e − iα ( c ) = X m ∈ Z d h α,n, m e − i h m ,k i , (2.25)where the coefficients h α,n, m have the form h α,n, m = X c ∈ e C n, m ω ( c ) e − iα ( c ) . (2.26)By the definition of the cycle index (2.9), for each cycle c of length n we have k τ ( c ) k P e ∈ c k τ ( e ) k nτ + , where τ + = max e ∈A ∗ k τ ( e ) k . Thus, h α,n, m = 0 for all m ∈ Z d such that k m k > nτ + .2.6. Proof of the main Theorems.
We present necessary and sufficient conditions underwhich the spectrum of the magnetic Schr¨odinger operators is flat.
Theorem 2.5.
Let H α = ∆ α + Q be the magnetic Schr¨odinger operator defined by (1.2) –(1.3) with a periodic magnetic potential α and a periodic electric potential Q on a periodicgraph G . Then the following statements are equivalent:(i) The spectrum of H α is flat.(ii) For each n ∈ N ν the trace of H nα ( k ) does not depend on k ∈ T d .(iii) The Fourier coefficients h α,n, m of Tr H nα ( k ) for all ( n, m ) ∈ N ν × (cid:0) Z d \ { } (cid:1) satisfy h α,n, m = X c ∈ e C n, m ω ( c ) e − iα ( c ) = 0 . (2.27) N SPECTRUM OF MAGNETIC SCHR ¨ODINGER OPERATORS ON PERIODIC GRAPHS 11 (iv) For any n ∈ N ν the following identities hold true π ) d Z T d Tr H nα ( k ) dk = | h α,n, | , where h α,n, = X c ∈ e C n, ω ( c ) e − iα ( c ) . (2.28) Here e C n, m is defined by (2.23); α ( c ) is the flux of the magnetic potential α through the cycle c defined by (2.9), and ω ( c ) is given by (2.20). Remark.
Theorem 2.5 also determines the necessary and sufficient conditions under whichthe spectrum of the magnetic Schr¨odinger operators has an a.c. component.
Proof.
The determinant of (cid:0) λI ν − H α ( k ) (cid:1) has the decompositiondet (cid:0) λI ν − H α ( k ) (cid:1) = ν Y j =1 ( λ − λ α,j ( k )) = λ ν + ξ λ ν − + ξ λ ν − + . . . + ξ ν − λ + ξ ν , (2.29)where the coefficients ξ j are given by (see, e.g., p. 87–88 in [3]) ξ n = − n (cid:18) T n + n − X j =1 T n − j ξ j (cid:19) , T n = Tr H nα ( k ) , n ∈ N ν , (2.30)and, in particular, ξ = − T , ξ = − ( T − T ) , . . . .( i ) ⇔ ( ii ). Let the spectrum of H α be flat. Then all band functions λ α,j ( · ), j ∈ N ν , areconstant, and, consequently, Tr H nα ( k ) = P νj =1 λ nα,j ( k ) does not depend on k for each n ∈ N ν .Conversely, let Tr H nα ( k ) does not depend on k for each n ∈ N ν . From this and (2.29),(2.30) it follows that the determinant det (cid:0) λI ν − H α ( k ) (cid:1) does not depend on k . Then all bandfunctions λ α,j ( · ), j ∈ N ν , are constant and all spectral bands σ j ( H α ) are degenerate.( ii ) ⇒ ( iii ). Let Tr H nα ( k ) do not depend on k for each n ∈ N ν . Then, using the Fourierseries (2.24), we obtain h α,n, m = X c ∈ e C n, m ω ( c ) e − iα ( c ) = 0 , ∀ ( n, m ) ∈ N ν × (cid:0) Z d \ { } (cid:1) . ( iii ) ⇒ ( ii ). Let the condition (2.27) hold true. Then, by Theorem 2.4, we obtainTr H nα ( k ) = h α,n, = X c ∈ e C n, ω ( c ) e − iα ( c ) , ∀ n ∈ N ν , (2.31)i.e., the traces Tr H nα ( k ), n ∈ N ν , do not depend on k .( iii ) ⇔ ( iv ). Using the Parseval’s identity for the Fourier series (2.24), we obtain1(2 π ) d Z T d Tr H nα ( k ) dk = X m ∈ Z d | h α,n, m | , ∀ n ∈ N . Thus, the condition (2.27) and (2.28) are equivalent.We prove Theorem 1.3 about the flat spectrum of the magnetic Schr¨odinger operators.
Proof of Theorem 1.3.
The first spectral band σ ( H ) of the Schr¨odinger operator H without magnetic field is non-degenerate (see Theorem 2.1. ii in [11]). Then the a.c. spectrumof H is not empty. Then, by Theorem 2.5, there exists ( n, m ) ∈ N ν × (cid:0) Z d \ { } (cid:1) such that the Fourier coefficient h ,n, m = 0, where h ,n, m is defined by (2.24). For this ( n, m ) and for themagnetic potential tα we define the function f ( t ) := h tα,n, m = X c ∈ e C n, m ω ( c ) e − itα ( c ) , t ∈ R , and note that the sum is finite. This function has an analytic extension to the whole complexplane. If f = const, then we obtain f = f (0) = h ,n, m = 0 for such specific ( n, m ). ThenTheorem 2.5 yields that the a.c. spectrum of the operator H tα is not empty for all t ∈ R .If f = const, then f has a finite number of zeros on any bounded interval. Then Theorem2.5 yields that the a.c. spectrum of H tα is not empty for all except finitely many t ∈ [0 , Acknowledgments.
Our study was supported by the RFBR grant No. 19-01-00094.
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