Gauge Equivalence and the Inverse Spectral Problem for the Magnetic Schrödinger Operator on the Torus
aa r X i v : . [ m a t h . SP ] D ec GAUGE EQUIVALENCE AND THE INVERSESPECTRAL PROBLEM FOR THE MAGNETICSCHR ¨ODINGER OPERATOR ON THE TORUS
G. ESKIN AND J. RALSTON,DEPARTMENT OF MATHEMATICS, UCLA,LOS ANGELES, CA 90095-1555, USA
In memory of Mark Iosifovich Vishik
Abstract.
We study the inverse spectral problem for the Schr¨odingeroperator H on the two-dimensional torus with even magnetic field B ( x ) and even electric potential V ( x ). V.Guillemin [11] provedthat the spectrum of H determines B ( x ) and V ( x ). A simpleproof of Guillemin’s results was given by the authors in [3]. Inthe present paper we consider gauge equivalent classes of magneticpotentials and give conditions which imply that the gauge equiva-lence class and the spectrum of H determine the magnetic field andthe electric potential. We also show that generically the spectrumand the magnetic field determine the “extended” gauge equivalenceclass of the magnetic potential. The proof is a modification of theproof in [3] with some corrections and clarifications. Introduction
Let L = { m e + m e : m = ( m , m ) ∈ Z } be a lattice in R .Here { e , e } is a basis in R . We assume that the lattice L has thefollowing property:(1.1) For d, d ′ ∈ L, if | d | = | d ′ | , then d ′ = ± d. Let L ∗ = { δ ∈ R : δ · d ∈ Z for all d ∈ L } be the dual lattice. Weconsider a Schr¨odinger operator of the form(1.2) H = (cid:16) − i ∂∂x − A ( x ) (cid:17) + (cid:16) − i ∂∂x − A ( x ) (cid:17) + V ( x ) , x ∈ R , where A ( x ) = ( A ( x ) , A ( x )) is the magnetic potential and V ( x ) is theelectric potential.Let B ( x ) be the magnetic field,(1.3) B ( x ) = curl A ( x ) = ∂A ∂x − ∂A ∂x . e assume that B ( x ) and V ( x ) are periodic, i.e. B ( x + d ) = B ( x ) , V ( x + d ) = V ( x ) , ∀ d ∈ L, i.e. B ( x ) and V ( x ) are smooth functions on T = R /L . We alsoassume that B ( x ) and V ( x ) are even, i.e. B ( − x ) = B ( x ) and V ( − x ) = V ( x ).Denote by G ( T ) the gauge group of complex-valued functions g ( x ) ∈ C ∞ ( T ) such that | g ( x ) | = 1. Any g ( x ) ∈ G ( T ) has the following form(1.4) g ( x ) = exp(2 πiδ · x + iϕ ( x )) , where δ ∈ L ∗ and ϕ ( x ) is periodic, ϕ ( x + d ) = ϕ ( x ) , ∀ d ∈ L . Theoperator of multiplication by g ( x ) transforms the equation Hu = λu to the equation H ′ u ′ ( x ) = λu ′ ( x ) , where H ′ has the form (1.2) with A ( x ) replaced by A ′ ( x ),(1.5) A ′ ( x ) = A ( x ) − ig − ( x ) ∇ g ( x ) = A ( x ) + 2 πδ + ∇ ϕ ( x ) , δ ∈ L ∗ , and u ′ ( x ) = g − ( x ) u ( x ). The magnetic potentials A ′ ( x ) and A ( x ) re-lated by (1.5) are called gauge equivalent. Since H and H ′ are unitarilyequivalent, they have the same spectrum.Let(1.6) B ( x ) = X β ∈ L ∗ b β e πiβ · x be the Fourier series expansion of B ( x ). We assume that the coefficient(1.7) b = | D | − Z D B ( x ) dx, is not zero. Here D is a fundamental domain for the lattice L given by(1.8) D = { t e + t e , | t j | ≤ , j = 1 , } and | D | is the area of D . Note that if x ∈ D , then − x ∈ D .Given B ( x ), we let(1.9) A ( x ) = A ( x ) + a + X β ∈ ( L ∗ \ a β e πiβ · x , where a = ( a , a ) is a constant,(1.10) A ( x ) = b − x , x )and(1.11) a β = b β (2 πi ) − ( β + β ) − ( − β , β ) . ote that curl A = B ( x ).Let A ′ ( x ) be any magnetic potential that is gauge equivalent to A ( x ).Since curl ∇ ϕ = 0, B ′ ( x ) = B ( x ) , where B ′ ( x ) = curl A ′ ( x ). Therefore A ′ ( x ) can be also represented inthe form (1.9) with a ′ not necessarily equal to a . For the gauge equiv-alence of A ′ ( x ) and A ( x ), in addition to the equality of the magneticfields, one needs (cf. (1.5))(1.12) a ′ − a = 2 πδ, for some δ ∈ L ∗ . The main question in this paper is: For the class ofmagnetic potentials considered here, to what extent do the spectra ofmagnetic Schr¨odinger operators and the gauge equivalence classes ofmagnetic potentials determine the magnetic and electric fields?In § H is a self-adjoint oper-ator with compact resolvent and hence has a discrete spectrum. Herewe will describe the gauge equivalence classes of the magnetic potentialassuming that the magnetic field B ( x ) is fixed.Let γ j , j = 1 , , be the basis of the homology group of the torus,given by γ j = { te j , ≤ t ≤ } . Let α j = Z γ j a · dx = a · e j , where a is the constant vector in (1.9). For any d = m e + m e ∈ L we have a · d = m α + m α , i.e. knowing { α , α } determines a · d for any d ∈ L .Let A ′ ( x ) be a magnetic potential of the form (1.9) with a replacedby a ′ . Define α ′ j = Z γ j a ′ · dx = a ′ · e j . Let { e ∗ , e ∗ } be the basis in L ∗ dual to { e , e } , i.e. e j · e ∗ k = δ jk .The potentials A ( x ) and A ′ ( x ) are gauge equivalent if and only ifcurl A = curl A ′ and (cf. (1.12))(1.13) α ′ j − α j = ( a ′ − a ) · e j = 2 πδ · e j , j = 1 , , for some δ ∈ L ∗ . Short equivalent forms of (1.13) are e iα j = e iα ′ j , j = 1 , , nd(1.14) e ia · d = e ia ′ · d for any d = n e + n e ∈ L. Changing x to − x we get the operator H ′ which is just H with a changed to a ′ = − a . Note that H and H ′ have the same spectrumbut their magnetic potentials are not gauge equivalent when a = 0.Since we are looking for consequences of isospectrality, we introduce aweaker notion of gauge equivalence, namely(1.15) cos a · d = cos a ′ · d, ∀ d ∈ L The condition (1.15) is equivalent to cos α j = cos α ′ j , j = 1 ,
2. Sincecos α j − cos α ′ j = 2 sin( α j − α ′ j α j + α ′ j , cos α j = cos α ′ j implies that either α j − α ′ j or α j − α ′ j is an integermultiple of 2 π . Thus there are two choices for each j : e ia · e j = e ia ′ · e j or e ia · e j = e − ia · e j . We will say that a ′ and a belong to the same“extended gauge equivalence class” if (1.15) holds. Thus for everyextended gauge equivalence class of magnetic potentials, there are fourchoices of a , including a ′ = a and a ′ = − a , giving distinct gaugeequivalence classes when a = 0.Our first result gives conditions for the spectrum of H and gaugeequivalence class of A to determine the fields: Theorem 1.1.
Let B ( x ) , V ( x ) be periodic and even smooth functions,and assume L satisfies the condition (1.1). Suppose (1.16) Z D B ( x ) dx = 2 π. Consider the spectrum of the Schr¨odinger operator H with A ( x ) havingthe form (1.9). Suppose that (1.17) | B ( x ) − b | < | b | , where b = 2 π/ | D | . Then the spectrum of H determines uniquely B ( x ) and V ( x ) assuming that cos a · d, d ∈ L, is given and (1.18) cos a · d = 0 for all d ∈ L. Theorem 1.2.
Assume that the conditions (1.1), (1.16), (1.17) hold,and that the spectrum of H and the magnetic field B ( x ) are given. If B ( x ) satisfies a generic condition (stated in the proof ), then cos a · d s determined for all d ∈ L , i.e. the exended gauge equivalence class of A ( x ) is determined by the spectrum and the magnetic field. It follows from Theorem 1.2 that if B ( x ) , V ( x ) are fixed and the ex-tended gauge equivalence classes of A ( x ) and A ′ ( x ) are different, i.e. ifcos a · d = cos a ′ · d for some d ∈ L , then the corresponding operators H and H ′ have different spectra. This confirms the Aharonov-Bohm effectstating that different gauge equivalence classes have a different quan-tum mechanical effects, for example, the corresponding Schr¨odingeroperators have different spectra(cf. [5]).The case when B ( x ) and V ( x ) are even and a = 0 (cf. (1.9)) wasproven in an important paper of Guillemin [11]. In [3] we reproved theresult of [6] by a different and simpler method.The method of the present paper is a modification of the method of[3] with some clarifications and corrections.We mention briefly some related results on the inverse spectral prob-lems in two and higher dimensions: The magnetic Schr¨odinger operatoron T with R D B ( x ) dx = 0 and A ( x ) periodic was studied in [1]. In[10] Gordon et al. generalized [11] to the case of n -dimensional tori.Guillemin and Kazhdan [13] studied the inverse spectral problem fornegatively curved manifolds. Guillemin [12] studied the inverse spec-tral problem on S . Zeldich [18] solved the inverse spectral problemfor analytic bi-axisymmetric plane domains. In [6] the inverse spectralproblems on the torus for the Schr¨odinger operator − ∆ + q ( x ) werestudied. See also [4],[8], [16].Gordon [7] and Gordon-Schuth [9] gave many interesting examplesof isospectral manifolds which were not isometric.2. The singularities of the wave trace
We introduce the “magnetic translation operators” (cf. [17])(2.1) T j u ( x ) = e − iA ( e j ) · x u ( x + e j ) , j = 1 , , where A ( x ) is from (1.10). These operators are required to commutewith each other and with H . This implies that(2.2) A ( e ) · e = − A ( e ) · e = πl, where l is an integer. Using (1.7), (1.10) we get that (2.2) is equivalentto(2.3) Z D B ( x ) dx dx = 2 πl. ater we shall assume that l = 1. Having that T , T and H commutewe denote by D the subspace of the Sobolev space H ( R ) consistingof u ( x ) ∈ H ( R ) such that T j u = u, j = 1 ,
2. Then the operator H is self-adjoint in L ( D ) on the restriction of D to the fundamentaldomain D . We shall denote this operator by H D .Let λ ≤ λ ≤ λ ≤ ... be the spectrum of H D . and let E D ( x, y, t )be the fundamental solution for the wave equation on R /L . Then thewave trace formula gives the equality as distributions in t (2.4) ∞ X j =1 cos t p λ j = Z D E D ( x, x, t ) dx. The distribution E D ( x, y, t ) is defined as follows: Let E ( x, y, t ) be thefundamental solution for the wave equation on R : ∂ E ( x, y, t ) ∂t + HE ( x, y, t ) = 0 , x ∈ R , y ∈ R , (2.5) E ( x, y,
0) = 0 , ∂E ( x, y, ∂t = δ ( x − y ) , x ∈ R , y ∈ R . Then(2.6) E D ( x, y, t ) = X ( m,n ) ∈ Z T m T n E ( x, y, t ) . Note that(2.7) T m T n E ( x, y, t ) = e − iA ( d ) · x E ( x + d, y, t ) , where d = me + ne . We used in (2.7) that A ( e j ) · e j = 0. Since E ( x, y, t ) is singular only when | x − y | = t and since the condition (1.1)holds, the singularities of the trace (2.4) at t = | d | , d = m e + m e come only from two terms(2.8) Z D ( T m T n E ( x, x, t ) + T − m T − n E ( x, x, t )) dx. To compute the singularities in (2.8) we will use as in [3] and [4] theHadamard-H¨ormander parametrix (cf. [14], [15]). We have(2.9) E ( x, y, t ) = ∂∂t ( E + ( x, y, t ) − E + ( x, y, − t ) , where E + ( x, y, t ) is the forward fundamental solution:(2.10) (cid:16) ∂ ∂t + H (cid:17) E + ( x, y, t ) = δ ( t ) δ ( x − y ) , E + ( x, y, t ) = 0 for t < . t follows from [14] that(2.11) E + ( x, y, t ) = m ( x, y ) 12 π ( t − | x − y | ) − + + m ( x, y )2 − π − ( t − ( x − y ) ) + + O (( t − ( x − y ) ) ) , where(2.12) m ( x, y ) = exp (cid:16) i Z ( x − y ) · A ( y + s ( x − y )) ds (cid:17) , (2.13) m ( x, y ) = − m ( x, y ) (cid:16) Z V ( y + s ( x − y )) ds + b ( x, y ) (cid:17) , where(2 . ′ ) b ( x, y ) = h(cid:16) − i ∂∂x − A ( x ) (cid:17) m i m − ( x, y ) . Let(2.14) I ( d ) = Z D exp i h − A ( d ) · x + Z ( A ( x + sd ) · d ) ds i dx (2.15) J ( d ) = Z D Z [( V ( x + sd )+ b ( x + sd, x )] ds exp h − iA ( d ) · x + i Z ( A ( x + sd ) · d ) ds i dx It follows from (2.8)-(2.13) that I ( d ) + I ( − d ) and J ( d ) + J ( − d ) aredetermined by the spectrum of H .Using that A ( d ) · d = 0 and that A ( d ) · x = − A ( x ) · d we canrewrite (2.14) in the form(2.16) I ( d ) = Z D exp i h A ( x ) · d + a · d + Z ( A ( x + sd ) · d ) ds i dx, where(2.17) A ( x ) = X β ∈ L ∗ \ a β e πiβ · x , and a β are defined in (1.11).Let { e ∗ , e ∗ } be the basis in L ∗ dual to the basis { e , e } , i.e.(2.18) e ∗ j · e k = δ jk , ≤ j, k ≤ . e shall construct e ∗ j , j = 1 , , explicitly. Let e ⊥ j = ( − e j , e j ), Denoteby ∆ the determinant (cid:12)(cid:12)(cid:12)(cid:12) e e e e (cid:12)(cid:12)(cid:12)(cid:12) . We assume that ∆ >
0. Note that ∆is the area of the fundamental domain D : ∆ = | D | . Now we define(2.19) e ∗ = − e ⊥ , e ∗ = 1∆ e ⊥ . We have d = me + ne = k ( m e + n e ), where k ≥
1, is an integerand m o , n have no common factors. Then A ( d ) = b d ⊥ = b k m e ⊥ + n e ⊥ ) = b k ∆2 ( m e ∗ − n e ∗ ) = b k ∆2 δ, where δ = − n e ∗ + m e ∗ .Using (1.7), (2.3) we get(2.20) A ( d ) = πklδ. If β · d = 0, then R e πisβ · d ds = 0. When β · d = 0, we have β = pδ, p ∈ Z \ O, δ = − n e ∗ + m e ∗ . We shall compute the inner product d · a pδ , where a β is given by (1.11). Note that d = k ( m e + n e ) , δ ⊥ = − n ( e ∗ ) ⊥ + m ( e ∗ ) ⊥ = ( n e + m e ) (cf. (2.19). Therefore d · a pδ = kb pδ p | m e + n e | ∆2 πip | δ ⊥ | = kb pδ ∆2 πip , since | δ ⊥ | = | m e + n e | . It follows from (1.7) and (2.3) that∆ = | D | = πlb . Hence(2.21) d · a pδ = klb pδ ipb . Therefore I ( d ) has now the form(2.22) I ( d ) = e ia · d Z D exp 2 πikl (( x · δ ) + A δ ( δ · x )] dx, where(2.23) A δ ( δ · x ) = X p =0 b pδ πipb e πip ( δ · x ) . Potentials of the form (2.23) are called “ directional potentials” in [6].Note that dds A δ ( s ) = 1 b B δ ( s ) , where(2.24) B δ ( s ) = X p =0 b pδ e ips s a directional potential for B ( x ). ¿From here on we assume that l = 1and set d = k d = m e + n e .Choose δ ′ ∈ L ∗ so that ( δ, δ ′ ) is a basis in L ∗ and let ( γ, γ ′ ) be thebasis in L dual to ( δ, δ ′ ) ∈ L ∗ . We let D ′ be the fundamental domain for L with respect to the basis { γ, γ ′ } in the form { sγ + s ′ γ ′ , − ≤ s, s ′ ≤ } , and continue to let D be the fundamental domain for L from (1.8).Setting x = sγ + s ′ γ ′ , we x · δ = s, x · δ ′ = s ′ . Since the image of D is D ′ , using this change of variables we get(2.25) Z D exp 2 πik (( x · δ )+ A δ ( x · δ )) dx = / Z − / / Z − / [exp 2 πik ( s + A δ ( s ))] c dsds ′ , where c = (cid:12)(cid:12) ∂ ( x ,x ) ∂ ( s,s ′ ) (cid:12)(cid:12) is the Jacobian. Note that c is the area of afundamental domain for L . Therefore, integrating in s ′ we get I ( d ) = c / Z − / exp h πik (cid:16) s + a · d π + A δ ( s ) (cid:17)i ds. When a = 0 and A δ ( s ) is an odd function of s then I ( − d ) = I ( d ) andthe computations are simplified. When a = 0 the spectral invariant is I ( d ) + I ( − d ) and we get, changing k to − k :(2.26) I ( d ) + I ( − d ) = 2 c / Z − / cos h πk (cid:16) s + a · d π + A δ ( s ) (cid:17)i ds Since / R − / e πi ( x + sd ) · δ ′ ds = 0 if d · δ ′ = 0 and it is equal e πix · δ when δ · d = 0, the directional potential B δ is given by B δ ( x ) = / Z − / ( B ( x + sd ) − b ) ds. Hence, by the assumption | B ( x ) − b | < | b | (2.27) max | B δ ( x ) | ≤ / Z − / max | B ( x ) − b | ds < | b | . Letting(2.28) y = s + A δ ( s ) , e have dyds = 1 + dds A δ ( x ) = 1 + B δ ( s ) b > . Therefore the inverse function s = s ( y ) , y ∈ R , is defined. Since y = y ( s ) is odd, the inverse function s = s ( y ) is also odd. Since y ( s + 1) = y ( s ) + 1 we have s ( y + 1) = s ( y ) + 1. Differentiating in y weget s ′ ( y +1) = s ′ ( y ) , i.e. s ′ ( y ) is periodic of period 1. The function s ′ ( y )is even since s ( y ) is odd. Let e ( y ) = s ( y ) − y . Since s ( y + 1) = s ( y ) + 1we get that e ( y + 1) = e ( y ) , i.e. e ( y ) is periodic and odd. Note that s ′ ( y ) = 1 + e ′ ( y ) . After the change of variables s = s ( y ) in (2.26), we have I ( d ) + I ( − d ) = 2 c / A δ (1 / Z − / A δ ( − / cos(2 πky + ka · d ) s ′ ( y ) dy. Since cos(2 πky + ka · d ) s ′ ( y ) is periodic and A δ (1 /
2) = A δ ( − /
2) weget(2.29) I ( d ) + I ( − d ) = 2 c / Z − / cos(2 πky + ka · d )) s ′ ( y ) dy. Since (sin 2 πky ) s ′ ( y ) is an odd function on ( − / , / c / Z − / (sin ka · d )(sin 2 πky ) s ′ ( y ) dy = 0 , and this implies(2.30) I ( d ) + I ( − d ) = 2 c / Z − / (cos ka · d )(cos 2 πky ) s ′ ( y ) dy As an even smooth function, s ′ ( y ) is a sum of its Fourier cosineseries on ( − / , / ka · d ) is known and nonzero forall k ≥
1. Then we know the Fourier cosine coefficients for k ≥
1. Thisuniquely determines s ′ ( y ) up to a constant. Therefore s ′ ( y ) = C + s ( y ),where / R − / s ( y ) dy = 0. Since s ( y ) = y + e ( y ), where e ( y ) is periodicand odd we get s ′ ( y ) = 1 + e ′ ( y ). Knowing s ′ ( y ) we can find s ( y )and subsequently A δ ( s ) from the knowledge of the spectrum of H and os( ka · d ) , k ≥
1. Repeating the same arguments for any d ∈ L wecan recover A ( x ).Now we can recover V ( x ) assuming that R D V ( x ) dx = 0. One cancheck from (2.14 ′ ) that b ( x, y ) does not depend on a : the only term in m ( x, y ) (cf. (2.12)) that contains a has the form e ia · ( x − y ) . Therefore − i ∂∂x m ( x, y ) = ( a + c ( x, y )) m ( x, y ) where c ( x, y ) is independentof a . Hence ( − i ∂∂x − A ) m ( x, y ) = ( a + c ( x, y ) − ( a + b x ⊥ + A ( x )) m ( x, y ) will not contain a . Therefore b ( x, y ) is known oncewe know A ( x ).Since R V ( x + sd ) ds = V δ ( x ), we have(2.31) J ( d ) = J ( d ) + J ( d ) , where(2.32) J ( d ) = Z D V δ ( x · δ ) exp[2 πik ( x · δ + a · d π + A δ ( x · δ ))] dx, and J ( d ) is the term containing b ( x, y ), i.e. J ( d ) is known. Therefore J ( d ) + J ( − d ) is determined by the spectrum assuming that cos ka · d is known.Making the change of variables x = sγ + s ′ γ ′ as in (2.25) and inte-grating in s ′ we get(2.33) J ( d ) + J ( − d ) = 2 c / Z − / V δ ( s ) cos 2 πk ( s + a · d π + A δ ( s )) ds. Making the change of variables y = s + A δ ( s ) as in (2.29) we get(2.34) J ( d ) + J ( − d ) = 2 c / Z − / V δ ( s ( y )) s ′ ( y ) cos(2 πky + ka · d ) dy, where s = s ( y ) is the inverse to y = s + A δ ( s ). Note that s ′ ( y ) is evenperiodic function of period 1, and V δ ( s ( y )) is also even periodic, since V ( x ) is even periodic and s ( y ) is an odd function satisfying s ( y + 1) = s ( y ) + 1. ince V δ ( s ( y )) s ′ ( y ) is an even function, R − V δ ( s ( y )) s ′ ( y ) sin 2 πkydy =0 . Thus, as in (2.30) we have:(2.35) J ( d ) + J ( − d ) = 2 c cos( ka · d ) / Z − / V δ ( s ( y )) s ′ ( y ) cos 2 πkydy, k ≥ . Knowing J ( d ) + J ( − d ) and cos( ka · d ) we know the Fourier cosinecoefficients of the even function V δ ( s ( y )) s ′ ( y ) for k ≥
1. Therefore wecan determine V δ ( s ( y )) s ′ ( y ) up to a constant(2.36) V δ ( s ( y )) s ′ ( y ) = C + e ( y ) , where e ( y ) is known and R / − / e ( y ) dy = 0.Integrating (2.36) we get C = / Z − / V δ ( s ( y )) s ′ ( y ) dy = / e (1 / Z − / e ( − / V δ ( s ) ds = / Z − / V δ ( s ) ds = 0 , where s ( y ) = y + e ( y ) , and e ( y ) is periodic with period 1. Thus we canrecover V δ ( s ) for each δ ∈ L ∗ and therefore we can recover V ( x ). Thisconcludes the proof of Theorem 1.1.Now we shall prove Theorem 1.2. We shall assume that the magneticfield is generic in the following sense: There are two directions δ and δ which form a basis for L ∗ such that the directional fields B δ ( s )and B δ ( s ) are not identically zero. In this case the functions s ( x )and s ( x ) (cf. (2.28)) corresponding to δ and δ respectively, are notidentically zero. We make the additional generic assumption that a j =0 , j = 1 , , where a kj are the Fourier cosine coefficients of s ′ j ( y ) , k ≥ a · d j , j = 1 , , where { d , d } is the dual basis to { δ , δ } . Given d ∈ L , there are areintegers m and n such that d = md + nd . Hencecos a · d = Re { e ia · ( md + nd ) } = Re { (cos( a · d ) ± i p − (cos( a · d ) ) m (cos( a · d ) ± i p − (cos( a · d ) ) n } , and, since the ± ’s disappear when one takes the real part, cos( a · d ) isdetermined by cos( a · d ) and cos( a · d ). Thus, cos a · d = cos a ′ · d for all d ∈ L as in (cf. (1.15)). This proves Theorem 1.2. eferences [1] G. Eskin, Inverse spectral problems for the Schr¨odinger equation with periodicvector potential, Comm. Math. Phys. 125(1989), 265-300[2] G. Eskin, A simple proof of magnetic and electric Aharonov-Bohm effect,Comm. Math. Phys., 32 (2013), 747-763[3] G. Eskin, J. Ralston, Remarks on spectral rigidity for magnetic Schr¨odingeroperators, Mark Krein Centenary Conference, vol. 2, 323-329, Operator TheoryAdv. Appl., 191, Birkhauser, Basel, 2009[4] G. Eskin, J. Ralston, Inverse spectral problems in rectangular domain, Comm.PDE, 32 (2007), 971-1000[5] G. Eskin, J. Ralston, The Aharonov-Bohm effect in spectral asymptotics of themagnetic Schr¨odinger operator, arXiv:1301.6217[6] G. Eskin, J. Ralston, E. Trubowitz, On isospectral periodic potentials in R n , Iand II, Comm. Pure and App. Math., 37 (1984), 647-676, 715-753[7] C. Gordon, Survey of isospectral manifolds, Handbook of Differential Geometry,Vol. 1, 747-778, North Holland, 2000.[8] C. Gordon, T. Kappeler, On isospectral potentials on tori, Duke Math. J. 63(1991), 217-233[9] C. Gordon, D. Sch¨uth, Isospectral potentials and conformally equivalentisospectral metrics on spheres, balls and Lie groups, J. Geometric Anal. 13(2003), 300-328.[10] C. Gordon, P. Guerini, T. Kappeler, D. Webb, Inverse spectral results on evendimensional tori, Annales de l’Institut Fourier, 58 (2008), 2445-2501[11] V. Guillemin, Inverse spectral results on two-dimensional tori, Journal of theAMS, 3 (1990),375-387[12] V. Guillemin, Spectral theory on S : some open questions, Adv. Math. 42(1981), 283-298[13] V. Guillemin, D. Kazdhan, Some inverse spectral results for negatively curved2-manifolds, Topology, 19 (1980), 301-312[14] J. Hadamard, Le probl`eme de Cauchy et les ´equations aux d´eriv´ees partielleslin´eaires hyperboliques, Hermann, Paris, 1932[15] L. H¨ormander, The Analysis of Linear Partial Differential Equations, III,Springer-Verlag, Vienna, 1985[16] A. Waters, Spectral rigidity for periodic Schr¨odinger operators in dimension 2,arXiv:1203:2901.[17] J. Zak, Dynamics of electrons in solids in external fields, Phys. Rev., 168(1968), 686-695[18] S. Zeldich, Spectral determination of analytic, bi-axisymmetirc plane domains,Geometric Funct. Anal. 10 (2000), 628-677: some open questions, Adv. Math. 42(1981), 283-298[13] V. Guillemin, D. Kazdhan, Some inverse spectral results for negatively curved2-manifolds, Topology, 19 (1980), 301-312[14] J. Hadamard, Le probl`eme de Cauchy et les ´equations aux d´eriv´ees partielleslin´eaires hyperboliques, Hermann, Paris, 1932[15] L. H¨ormander, The Analysis of Linear Partial Differential Equations, III,Springer-Verlag, Vienna, 1985[16] A. Waters, Spectral rigidity for periodic Schr¨odinger operators in dimension 2,arXiv:1203:2901.[17] J. Zak, Dynamics of electrons in solids in external fields, Phys. Rev., 168(1968), 686-695[18] S. Zeldich, Spectral determination of analytic, bi-axisymmetirc plane domains,Geometric Funct. Anal. 10 (2000), 628-677