We consider the Landau Hamiltonian , self-adjoint in , whose spectrum consists of an arithmetic progression of infinitely degenerate positive eigenvalues , . We perturb by a non-local potential written as a bounded pseudo-differential operator with real-valued Weyl symbol , such that is compact. We study the spectral properties of the perturbed operator . First, we construct symbols , possessing a suitable symmetry, such that the operator admits an explicit eigenbasis in , and calculate the corresponding eigenvalues. Moreover, for which are not supposed to have this symmetry, we study the asymptotic distribution of the eigenvalues of adjoining any given . We find that the effective Hamiltonian in this context is the Toeplitz operator , where is the orthogonal projection onto , and investigate its spectral asymptotics.
We consider the Landau Hamiltonian , i.e. the 2D Schrödinger operator with constant scalar magnetic field , self-adjoint in . We have
As is well known, the spectrum of the operator consists of eigenvalues of infinite multiplicity
called the Landau levels (see [21,30]).
Let be a bounded pseudo-differential operator (ΨDO) with real-valued Weyl symbol ; then is self-adjoint in . We assume moreover that is relatively compact with respect to , i.e. that the operator is compact. Proposition 3.3 below contains simple sufficient conditions which guarantee the validity of these general assumptions on . We will study the spectral properties of the perturbed operator
By the Weyl theorem on the invariance of the essential spectrum under relatively compact perturbations, we have
We will be interested, in particular, in the asymptotics of the discrete spectrum of near any fixed Landau level , . As we will see, generically, the effective Hamiltonian which governs this asymptotic behavior is the Toeplitz-type operator
considered as an operator in , where is the orthogonal projection onto .
Let us explain briefly our motivation to study the spectral properties of the operator . The so called non-local potentials defined as appropriate integral operators play an important role in nuclear physics (see e.g. [17,18,42]). We recall that any integral operator in , which has a reasonable integral kernel can be represented as a Weyl ΨDO (see (2.2) below). That is why, in the mathematical physics literature there is a persistent interest in Schrödinger operators with non-local, in particular, pseudo-differential potentials (see [2,19,28]).
Further, the authors of [10,12,44] investigate the spectral properties of Schrödinger operators with non-local potentials, and compare them to their homologues with appropriate effective local ones. The leading examples of the Hamiltonians considered in these articles can be written as
with real-valued rapidly decaying symbol (see, in particular, [12, Eq. (10)] and [10, Eq. (3)]), so that the complete Weyl symbol of the operator in (1.4) is
The spectral properties of these quantum Hamiltonians are investigated in [10,12,44] in the absence of exterior magnetic field. However, the influence of such a field is of primary interest in physics; for example, the discovery of the Zeeman effect of splitting of the atomic spectra in constant magnetic field, brought to Pieter Zeeman the 1902 Nobel prize in physics. By the so called minimal coupling principle1
This principle was criticized by the authors of [32] who suggested instead the so called magnetic Weyl calculus. However, if the magnetic field is constant and the magnetic potential is chosen to be linear, the two procedures give the same result.
, the presence of a constant magnetic field can be taken into account if we replace the symbol on (1.5) by
where is a magnetic potential which generates the magnetic field; in our case where , we can choose , . Setting , we obtain
which, if , is exactly the symbol of the operator in (1.2).
Note that articles [10,12,44] concern mostly the case . However, the spectrum of the 3D Schrödinger operator with constant magnetic field is purely absolutely continuous, so that the spectral properties of its perturbations are quite different than in the 2D case, and, in our opinion, should be examined separately. On the other hand, our experience makes us believe that in order to study successfully various interesting 3D phenomena like the threshold singularities of the spectral shift function or the accumulation of resonances at the Landau levels (see e.g. [5,6,16] in the case of local potentials), one should understand well the properties of the 2D operator investigated in this article.
Another side of our motivation to consider the operator is the fact that during the last three decades there have been published numerous works on the spectral asymptotics for various types of perturbations of . For example:
Electric perturbations, i.e. perturbations of by an additive real multiplier V which plays the role of an electric potential, were considered in [20,26,37,38];
Magnetic perturbations, i.e. perturbations of the constant magnetic field b by a variable one , which involve a first-order differential operator, were investigated in [26,41];
Metric perturbations, i.e. perturbations of the Euclidean metric by a variable metric , which involve a second-order operator, were studied in [26,31].
The perturbations, i.e. the quantities V, or g, considered in [26,37] are of power-like decay at infinity, while those studied in [20,31,38,41] are of exponential decay or compact support. Recently, several articles, [22,33,36], treated the eigenvalue asymptotics for the Landau Hamiltonian defined on the complement of a compact in , and equipped with Dirichlet, Neumann, or Robin boundary conditions. In this geometric setting, the effective Hamiltonian which governs the eigenvalue asymptotics near the Landau level is an integral operator sandwiched between the projections , quite similar to the Toeplitz operator in (1.3).
All these reasons are the source of our motivation to think of a unified approach to the spectral theory of pseudo-differential perturbations of magnetic quantum Hamiltonians. We believe that our present work could be a small but useful step in this direction.
Let us discuss briefly and informally the main results of the article. As already mentioned, most of them concern the eigenvalue distribution of the discrete eigenvalues of near the Landau levels. In particular, we compare the characteristic features of the eigenvalue asymptotics for the operator with non-local potential , and for with local potential with ; note that if V is local, then .
If is bounded, is compact, and , then the discrete eigenvalues of (resp., of ) may accumulate at any , , only from above (resp., only from below). However, in contrast to local symbols V, generally speaking, does not imply , and does not imply . Put
Suppose that
set
and denote by (resp., by ) the non-increasing (resp., non-decreasing) set of the eigenvalues of (resp., ), lying on the interval (resp., on ), .
If is a local potential of a definite sign, which decays at infinity and is sufficiently regular, then all the Landau levels, , are accumulation points of the discrete spectrum of , with a maximal admissible accumulation rate. More precisely, if the local potential does not vanish identically and satisfies, say, , and , then the results of [38] imply for every , i.e. the discrete eigenvalues of accumulate at each Landau level , and we have
i.e. the eigenvalues of cannot accumulate arbitrarily fast at , even if V has a compact support. If we assume in addition that , the distance between any two consecutive Landau levels, then it follows from the results of [29] that , i.e. the perturbations transform all the infinite-dimensional subspaces into trivial subspaces .
In contrast to these properties of with local decaying sign-definite V, we construct in Proposition 5.1 a Schwartz-class symbol such that , and the number of the eigenvalues of , lying on the interval , , and counted with the multiplicities, is equal to any given . If , then , i.e. the Landau level remains an eigenvalue of of infinite multiplicity. If , then the accumulation of at can be arbitrarily fast, i.e. there exists no maximal accumulation rate as in (1.7).
Further, as mentioned above, one of the main problems dealt with in the physics literature on (non-magnetic) Schrödinger operators with non-local potentials, is the existence of an effective local potential V which can replace the non-local one in a given asymptotic regime (see e.g. [10,12,44]). Let us describe briefly our construction which is similar to such a replacement. As already mentioned, we prove that the effective Hamiltonian which governs the eigenvalue asymptotics of with near is the Toeplitz operator defined in (1.3) (see the proof of Proposition 5.5). In Corollary 3.7, we show that is unitarily equivalent to the operator , compact in , whose symbol is a suitable integral transform of the symbol (see (3.41)). Next, we make the crucial assumption that the operator admits an anti-Wick symbol i.e. that is the convolution of with a Gaussian function (see (2.8)). Then is unitarily equivalent to the Toeplitz operator with
(see Corollary 3.8). Thus, could be regarded as the effective local counterpart of in the asymptotic analysis of the eigenvalue distribution near , , for the operator . We assume further that there exist and which does not vanish identically and has a compact support or decays exponentially at infinity, such that
where is the identity, and with is a partial differential operator of order (see (3.48)). The passage from to is motivated by the fact that might be non-negative even if does not have a definite sign. Then it follows from [8, Section 9], that is unitarily equivalent to the Toeplitz operator . Using this fact, we obtain several asymptotic terms of as for compactly supported (see Theorem 5.2), and for of exponentially decaying (see Theorem 5.3). Due to the fast decay of , the standard pseudo-differential techniques are not applicable and that is why we use and develop the methods of [20,31,38]. Finally, we drop our assumption that has a definite sign but assume that has a power-like decay at infinity, and in Theorem 5.4 we obtain the main asymptotic term of the local eigenvalue counting function as the energy approaches the Landau level , . Here, many traditional ΨDO techniques are applicable, in particular, we use and extend the methods developed in [37] and [13].
The article is organized as follows. In Section 2 we summarize the necessary facts from the general theory of ΨDOs with Weyl and anti-Wick symbols. Section 3 contains the description of several unitary operators which map to operators which are more accessible and easier to investigate. In particular, we show that , , is unitarily equivalent to . In Section 4 we deal with Weyl and anti-Wick ΨDOs with radial symbols, and obtain explicit formulas for their eigenvalues and eigenfunctions. Some of these results are known, others, to our best knowledge, are new and could be of independent interest. As a corollary, we construct a family of symbols , possessing a suitable symmetry such that the operator has explicit eigenvalues and orthonormal basis of eigenfunctions. In Section 5 we consider the eigenvalue distribution near the Landau level , , for the operator . First, in Proposition 5.1 we construct our explicit example of a symbol in the Schwartz class such that , which shows that the asymptotic behavior of the discrete spectrum of the operator near a given could be arbitrarily fast in contrast to the case of a local potential V. Next, we examine the eigenvalue asymptotics for the operators with , assuming that the operator admits an anti-Wick symbol related to through (1.8) and (1.9), and that decays exponentially at infinity or has a compact support. Finally, Theorem 5.9 contains our result on the eigenvalue asymptotics for near the Landau level , , in the case where is not supposed to have a definite sign but has a power-like decay at infinity.
Weyl and anti-Wick ΨDOs
In this section we recall briefly some basic facts from the theory of ΨDOs with Weyl and anti-Wick symbols, assuming that the dimension . We will use the following notations. Let X be a separable Hilbert space with scalar product , linear with respect to the first factor, and norm . By (resp., by ) we will denote the space of linear bounded (resp., compact) operators in X, and by , , the pth Schatten–von Neumann space of operators ) for which the norm is finite. In particular, is the trace class, and is the Hilbert–Schmidt class.
Let be the Schwartz class over , and be its dual class. For we define the ΨDO with Weyl symbol as the operator with integral kernel
Note that if a given linear operator T acting in has an integral kernel, say, , then T can be written as a ΨDO with Weyl symbol
(see e.g. [43, Eq. (23.39)]).
Let . Define the Wigner transform of the pair by
Then and we have . Moreover, the Wigner transform extends to in which case
By (2.2), the function coincides with the Weyl symbol of the operator with integral kernel , . Note that if and , then
Therefore, if , then (2.3) defines a linear continuous mapping .
Let us now introduce the Fourier transform
for , , and then extend it to . In particular, Φ extends to a unitary operator in .
If , then the integral kernel of the operator can be written not only as in (2.1) but also as
Let , , denote the set of functions such that
Note that .
Let. Thenextends to an operator bounded in. Moreover, there exists a constantindependent of, such that
We will consider Weyl ΨDOs acting in , under the generic assumption ; then, by Proposition 2.1, we have . However, many assertions in the sequel remain valid under more general assumptions about .
Further, for and , introduce the Hörmander–Shubin class
Let,, be a linear symplectic transformation. Assume that. Then there exists a unitary operatorsuch that
Proposition 2.4 remains valid for a considerably wider class of symbols including the linear and the quadratic ones.
The operator is called the metaplectic operator generated by the linear symplectomorphism κ.
Further, we discuss the anti-Wick ΨDOs. Let at first . Set
and define the operator
Then we will say that is a ΨDO with anti-Wick symbol . If and , then, similarly to (2.3), we have
where is the Husimi transform of . Therefore, if , then (2.9) defines, similarly to (2.3), a linear continuous mapping .
Since the convolution with the Gaussian function may improve the regularity and the decay rate of the symbol , the definition of the anti-Wick ΨDOs can be extended to a class of symbols, considerably larger than . In particular, we have the following
Set
Note that if , then . Our generic assumption concerning anti-Wick ΨDOs will be . As in the case of Weyl ΨDOs, many assertions in the sequel hold true under wider assumptions.
Let us give an alternative definition of the anti-Wick ΨDO with . For set
and introduce the rank-one orthogonal projection
Then we have
where the integral is understood in the weak sense. Identity (2.10) implies the monotonicity of with respect to the symbol . Namely, we have the following important
Assume that, andfor almost every. Then.
Not every Weyl ΨDO admits an anti-Wick symbol . If is a given Weyl symbol, then in order to find the corresponding anti-Wick symbol we have to solve the equation
i.e. to invert the so called Weierstrass transform, or, which is equivalent, to solve the inverse heat equation (see [43, Remark 24.2]). For example, if and , then there exists no such that (2.11) holds true. On the other hand, if the Fourier transform of is in , then admits an anti-Wick symbol given by
Unitary equivalences of the operators
In this section we establish unitary equivalences for the Landau Hamiltonian and its perturbation . First, we describe a suitable spectral representation of .
Let , so that . Introduce the magnetic creation operator
and the magnetic annihilation operator
The operators a and are closed on , and are mutually adjoint in . Moreover,
and
Therefore,
and, by (3.2), we have
Up to the unitary mapping , coincides with the Fock–Segal–Bargmann space of holomorphic functions (see e.g. [24, Section 3.2]).
Next, we recall that is unitarily equivalent under an appropriate metaplectic mapping to the operator , where
is the harmonic oscillator, self-adjoint in , and essentially self-adjoint on , while is the identity in . Let us recall the spectral properties of . We have
where
are the standard annihilation and creation operators which are closed on , and are mutually adjoint in . Moreover, they satisfy the commutation relation
Therefore,
Since
we have
Denote by the orthogonal projection onto , . Set
Then the functions satisfy
and form an orthogonal eigenbasis in . A simple calculation shows that
where
is the Hermite polynomial of degree q (see e.g. [23]). Thus, the functions , , form an orthonormal basis in . Introduce the Wigner functions
If , we will write
Lemma 3.1 below contains explicit expressions for , . In order to formulate it, we introduce the (generalized) Laguerre polynomials
As usual, we will write
Let. Then forwe haveIn particular,whereis a symmetric real valued matrix. Moreover,
An elementary calculation taking into account the parity of the Hermite polynomials easily yields
Changing the variable , and applying a standard complex-analysis argument showing that we can replace the interval of integration by , we find that
By [23, Eq. (7.377)],
Putting together (3.15), (3.16), and (3.17), we obtain (3.12). □
By (3.14) with , we have
where is the Gaussian defined in (2.7).
For , , set
Evidently, the mapping is linear and symplectic. Introduce the Weyl symbol
of the operator defined in (1.1). Then we have
Note that the function on the r.h.s. of (3.21) coincides with the Weyl symbol of the operator . Next, define the unitary operator by
where
Writing as a product of elementary linear symplectic transformations (see e.g. [25, Lemma 18.5.8]), and composing the corresponding elementary metaplectic operators, we easily check that is a metaplectic operator generated by the symplectic mapping in (3.19).
We haveMoreover, if, thenwhere
Relation (3.23) (resp., (3.25)) follows from Proposition 2.4 and the remark after it, and (3.21) (resp., (3.26)). Similarly, relations (3.24) follow from Proposition 2.4 and the fact that the Weyl symbol of the operator a (resp., ) is mapped under the symplectic transformation into the symbol of the operator (resp., ). □
Assume thatwith, anduniformly with respect to the variables on. Then,is bounded, andis compact in.
Since , the boundedness of follows from Proposition 2.1. By Proposition 3.2, we have
By the pseudo-differential calculus, we easily find that the Weyl symbol of the operator is in the class , while (3.27) guarantees that this symbol decays at infinity. Then Proposition 2.2 implies that , and by (3.28) we find that the operator is compact as well. □
Our next goal is to establish the unitary equivalence between and an operator . Similarly, we will establish the unitary equivalence between the Toeplitz operator with fixed , defined in (1.3), and an operator . To this end, we need the canonical basis of , . Let at first . Then the functions
form a natural orthogonal basis of (see e.g. [24, Sections 3.1–3.2]). Normalizing, we obtain the following orthonormal basis of :
Let now . Set
The commutation relation (3.3) easily implies
Therefore, the functions
form an orthonormal basis of , .
The functions admit an explicit expression, namely
the Laguerre polynomials being defined in (3.10).
Let . Set
The facts that is an orthonormal basis in , while is an orthonormal basis of with fixed , imply immediately the following elementary
Let.
The operatoris unitarily equivalent todefined by
Fix. Then the operatoris unitarily equivalent todefined by
We would like to give a more explicit form of the matrices defining the operators and , . To this end we need the following important
We havewhere,, are the Hermite functions defined in (
3.7
).
By (3.29)–(3.31), and (3.24), we get
where
Using (3.22), we easily find that
where
An elementary calculation yields
Inserting (3.38) into (3.37), we get
and inserting (3.39) into (3.36), we obtain (3.35). □
By (3.35), we have
Let . Set
where is the symbol defined in (3.26), and is the Wigner function defined in (3.9).
At the end of this section we consider the important case where the operator admits an anti-Wick symbol . Set
Then, of course, .
Let,. Assume that the operatorhas an anti-Wick symbol. Then,whereis the unitary operator defined in (
3.44
), andis the symbol defined in (
3.46
).
Assume at first that . Then, by Corollary 3.7, the operator is unitarily equivalent under the operator to a ΨDO with Weyl symbol
where we have taken into account (3.18). Thus we get (3.47) for . The result for general is obtained by an approximation argument similar to the one applied in the proof of [35, Theorem 2.11]. □
The operator admits a further useful unitary equivalence. For set
where is the Laguerre polynomial defined in (3.11). Thus, if , we have , and if , then is a partial differential operator with constant coefficients of order .
Assume that, and there exist,such thatThen the operatoris unitarily equivalent to the operator.
By [8, Lemma 9.2] and (3.49), we have
Let . Then with . Define the unitary operator by . Then (3.50) implies that
□
Spectral properties of Weyl ΨDOs with radial symbols
In this section we recall the fact that the Weyl ΨDOs with radial symmetric symbols are diagonalizable in the basis formed by Hermite functions, and obtain explicit expressions for the eigenvalues of the operators and .
Let . We will say that the symbol is radial if there exists a function with , such that
We will say that is radial if for each there exists a radial symbol such that
being the usual pairing between and . Note that if is radial, then its Fourier transform is radial as well. Moreover, if the radial symbol is real-valued, then is real-valued as well. Set
As is well known, , and hence , are orthonormal bases in . Note that the corresponding Fourier coefficients are defined not only for functions in but also for elements of , as well as for more general distributions (see e.g. [15,27]).
Letbe a radial symbol. Then the operatorhas eigenfunctionswithbeing the Hermite functions defined in (
3.7
), which correspond to eigenvalues
Letbe a radial anti-Wick symbol. Then the eigenfunctionsof the operatorcorrespond to eigenvalues
In view of (2.4), it is not unnatural to express the eigenvalues of in terms of the Fourier transform of the symbol , as in (4.2).
We have
Due to the radial symmetry of and (3.13), we find that
By (3.14),
Changing the variables , , and then , , we obtain (4.1). In order to check (4.2), we first note that by the Parseval identity,
By [35, Eq. (3.6)], we have
Therefore,
which implies (4.2). Let us now handle the anti-Wick case. Similarly to (4.4)–(4.5), we have
A simple calculation yields
where
The function g extends to an entire function satisfying
Using the first representation of the Laguerre polynomials in (3.11), we get
Inserting (4.9) into (4.8), changing the variables , and then inserting (4.8) into (4.7), we get (4.3). □
In view of (4.1), (2.8) and (3.14), relation (4.3) is equivalent to the fact that the Husimi function can be written as
Probably, (4.10) is known to the experts but since we could not find it in the literature, we include a proof of (4.3).
Letbe a radial symbol.
Thenif and only if the Fourier coefficients of the function,, with respect to the system, are non-negative.
Equivalently, we haveif and only if the Fourier coefficients of the function,, with respect to the systemare non-negative.
The first part follows from (4.1), and the second one from (4.2). □
The criterion in the first part of Corollary 4.2 has been established in [46] for the one-dimensional case , and in [27] for the multidimensional case. Presumably, at heuristic level, these facts have been known since long ago.
In the case , Proposition 4.1 tells us that the matrix is diagonal, provided that the symbol is radial. This fact admits an obvious generalization to the case where has a finite Fourier series with respect to the angle θ.
Let. Assume that there existssuch thatThen the matrixis-diagonal.
Of course, Proposition 4.4 admits an easy extension to any dimension .
Proposition 4.1 allows us to calculate explicitly the spectrum of the perturbed Landau Hamiltonian provided that the symbol is radial.
Let. Assume that the symbolis radial. Then the operator, normal on the domain, has eigenfunctionswhich correspond to eigenvalues
By Proposition 3.2 we have
while Lemma 3.5 and Proposition 4.1 imply
□
Eigenvalue distribution for the operator
Main results
In this section we study the eigenvalue asymptotics near a fixed Landau level , , of the perturbed Landau Hamiltonian with appropriate symbol such that is bounded, self-adjoint in , and relatively compact with respect to .
Proposition 5.1 below shows, in particular, that the eigenvalues of with and , adjoining the Landau levels , , may have quite arbitrary asymptotic behavior; they may not accumulate at a given , or may accumulate at any prescribed sufficiently fast accumulation rate.
Let T be an operator, self-adjoint in a given Hilbert space, and be an open interval with . Set
Here and in the sequel denotes the characteristic function of the set S. Thus, is just the spectral projection of T corresponding to the interval . If , then is the number of the eigenvalues of T, lying on and counted with the multiplicities. If, moreover, , then .
Letbe a given sequence with,. Then there exists a symbolsuch thatis radial,, andwhereare the intervals defined in (
1.6
).
Set
If , it suffices to take . Assume . Let be a decreasing set of numbers ; if , we can omit the condition . If , we assume that for any . Fix . Let be a decreasing set of numbers . If , we assume that for any . Now put
Then, (see [15, Theorem 2.5 (a)]), and, evidently, is radial. Moreover, by Corollary 4.5, and
By construction, all the eigenvalues , , lying in with , are simple. Therefore, (5.1) holds true. □
The proof of Proposition 5.1 contains an explicit construction of a negative compact perturbation of so that the eigenvalues may accumulate to only from below. Of course, it is possible to construct positive compact perturbations whose eigenvalues may accumulate to only from above, or self-adjoint compact perturbation with non-trivial positive and negative parts whose eigenvalues may accumulate to both from above and from below.
It is easy to check that if for some we have , then the Landau level remains an eigenvalue of infinite multiplicity of . In contrast to this situation, it was shown in [29] that if is local, i.e. if , , and , , then .
It is an elementary fact that if , then the operator
is bounded in , , the eigenvalues of T coincide with the set , while the eigenvalues of coincide with . Choosing appropriately the sequence c, we can easily obtain operators with various spectral properties. If, for example, is the set of the rational numbers on , and
then the operator will have purely dense point spectrum . Of course, in this case T is not relatively compact with respect to .
However, in the general case it would not be possible to interpret T as a Weyl ΨDO with a regular symbol. Our assumption in Proposition 5.1 that the sequence decays rapidly implies that the symbol defined in (5.2) belongs to the class . If, for example, we assume instead that we have only
then the symbol defined in (5.2) generates by Proposition 2.3 a Hilbert–Schmidt operator. In this case, (5.3) still holds true, just the eigenvalues of lying in a given gap may accumulate more slowly to .
In the following two theorems we assume that the operator satisfies two general assumptions:
The operator is bounded and self-adjoint, and the operator is compact in . Moreover, .
Let . Then the operator , being defined in (3.41), has an anti-Wick symbol . Moreover, there exists and such that (1.9) holds true, i.e we have
where is the symbol defined in (3.46), and is the differential operator defined in (3.48).
In what follows we will write ζ instead of .
It is easy to check that for each there exist symbols satisfying Assumptions and . A simple example can be constructed as follows. Pick , bounded together with all its derivatives. Set , , , , and
Then, according to (3.41), we have , and hence satisfies and . However, if we consider the operator defined in (5.2), and assume that for a certain we have , then the corresponding does not admit an anti-Wick symbol . Indeed, in this case we have
If with , then (4.6) easily implies that the Fourier transform of is a polynomial so that with .
As we will see in the proof of Theorem 5.5 below, the Toeplitz operator is the effective Hamiltonian which governs the eigenvalue asymptotics of near the Landau level , . The operator is an appropriate restriction of non-local ΨDO , and is unitarily equivalent by Corollary 3.7 to . By our assumption, admits an anti-Wick symbol and, hence, by Corollary 3.8 it is unitarily equivalent to , a restriction of the local multiplier . Thus, the existence of an anti-Wick symbol of allows us to replace, in a certain sense, the non-local operator by the local one in the asymptotic analysis of the eigenvalue distribution of near . As mentioned in the Introduction, similar substitutions of non-local potentials by local ones have been considered in the physics literature (see e.g. [10 ,12 ,44]).
We introduce the passage from to ζ in (1.9) in particular due to our requirement that ζ is non-negative: it may happen that while is not sign-definite.
In Theorem 5.2 (resp., Theorem 5.3) below we study the eigenvalue asymptotics for the operators at a given Landau level , , under Assumptions and , supposing in addition that ζ is compactly supported (resp., that ζ decays exponentially at infinity). Since by Assumption , the eigenvalues of the operator may accumulate to given Landau level only from above, while the eigenvalues of may accumulate to only from below, as mentioned in the Introduction. We recall the notations for the eigenvalues of , lying on , .
For the formulation of our first theorem we need the notion of a logarithmic capacity of a compact set (see e.g. [39, Chapter 5]). Let denote the set of probability measures on K. Then we have where
If , then, evidently, .
Letandwith fixed, hold true. Assume that,whereis a bounded domain with Lipschitz boundary, andon Ω. Then, and we have
Assume that , is compact and for some , , and we have where . Then [38] implies
which is a less precise version of (5.5).
By [45], Theorem 5.2 is valid under more general assumptions on . Namely, we can suppose that there exists a compact set such that only on and not on the entire domain Ω. We omit the details of the proof of this extension for the sake of the simplicity of the exposition.
Our next theorem concerns the case where ζ decays exponentially at infinity. Now we assume that and there exist and such that
uniformly with respect to . Set where is the constant scalar magnetic field.
Letandwith fixed, hold true. Assume that ζ satisfies (
5.7
). Thenand we have:
If, then there exist constants,, with, such that
If, then
If, then there exist constants,, such that
The coefficients and , , appearing in (5.8) and (5.10), are described explicitly in [31, Theorem 2.2]. For the completeness of the exposition, we reproduce this description here. Assume at first . For and , , introduce the function
Denote by the unique positive solution of the equation , so that . Set
Note that f is a real analytic function for small . Then , .
Let now . For and , , introduce the function
Denote by the unique positive solution of the equation so that . Define
which is a real analytic function for small . Then , .
In our next theorem we deal with the case where admits a power-like decay at infinity. Our general assumption concerning the perturbation is:
The symbol is real-valued and satisfies the hypotheses of Proposition 3.3.
We recall that under Assumption the operator is self-adjoint and bounded in , and is compact. However, we do not suppose now that has a definite sign. Further, under Assumption , there exists a symbol such that . By analogy with (3.41), set
Our next assumption concerns the decay of the symbols and at infinity:
Let . Then there exist and such that and .
A simple sufficient condition which guarantees the fulfillment of and is that with some and . In this case, the operator is not only bounded but also compact. Another condition which implies the validity of and is that is a local potential, and . This case corresponds to an electric perturbation of and was considered in [26,37].
It is more convenient to formulate Theorem 5.4 below in the terms of eigenvalue counting functions. For , , set
Let be a non-increasing function. We will say that f satisfies the condition if there exists such that:
f is derivable on ;
there exist numbers such that for any we have
Let . For a Lebesgue-measurable function set
where is the Lebesgue measure in .
Assume thatsatisfiesandwithand. Assume that the functions,being defined in (
3.41
), satisfy the condition. If(resp., if), then we haveor, respectively,
It is easy to show that there exists such that we can replace by in the remainder estimates in (5.12)–(5.13). Since anyway these remainder estimates would not be sharp, we omit the tedious technical details.
Let be a compact operator in a Hilbert space X. For set
If which is equivalent to , denote by the non-increasing sequence of the positive eigenvalues of T. If and , , then the Weyl inequalities
hold true (see e.g. [3, Theorem 9, Section 9.2]).
Suppose thatandwith, hold true and ζ satisfies the assumptions of Theorem
5.2
or of Theorem
5.3
. Then,, and for eachthere existssuch that for sufficiently largewe have
By the generalized Birman–Schwinger principle (see e.g. [1, Theorem 1.3], [4, Proposition 1.6]),
Writing
bearing in mind that the operator admits a uniform limit as , and applying the Weyl inequalities (5.14), we easily find that for each we have
as . Further, by Corollaries 3.7, 3.8, and 3.9, we have
Putting together (5.16), (5.17), and (5.18), we get
By [38], tends to infinity as which implies . By (5.19), the counting function also tends to infinity as , and hence . Finally, estimate (5.15) follows easily from (5.19). □
Let be a Jordan curve, i.e. a simple closed curve. We will call it -smooth if there exists a -smooth diffeomorphism .
Letbe a bounded domain. Then there exists a sequence of-smooth Jordan curvessuch that
We follow closely the main idea suggested by [40]. Let be a compact set, and . Set
Then [34, Satz 4] implies that if, in addition, K is connected, we have
By the left-hand inequality in (5.21), there exist in such that
Choosing points sufficiently close to we can find such that
Then there exists a -smooth Jordan curve
such that , . In order to see this, it suffices to connect the points , , by a piecewise-linear Jordan curve lying in Ω, and then smooth out the corners. Since is compact and connected, (5.22) implies
On the other hand, the right-hand inequality of (5.21) implies
Combining (5.24) and (5.25), we get
and, therefore,
Since , we have which together with (5.26) yields (5.20). □
Letbe a bounded domain. Then there exists a sequence of domainswith Lipschitz boundaries, such thatand
Let be the -smooth Jordan curve introduced in (5.23), and let , , be a normal unit vector to . Set
where is so small that and is Lipschitz-smooth. Since , (5.27) follows from (5.20). □
Pick a sequence of domains with Lipschitz boundaries such that and (5.27) holds true; the existence of such a sequence is guaranteed by Corollary 5.7. Set
Evidently, . Moreover,
By the mini-max principle, estimate (5.29) implies
By (5.28) and (5.30), we get
for every j. Combining (5.31) and (5.27), we obtain
Now (5.5) follows from (5.15) and (5.32).
For , , , and , set
where , , are the parameters introduced in the statement of the theorem. Arguing as in the proof of [31, Theorem 2.2], we can show that there exist , , and , such that
Since the functions are radial, we easily check that the eigenvalues of the operator coincide with the numbers
with and . Applying (5.33) and [31, Lemma 5.3], we find that
as , the coefficients and being introduced in the statement of Theorem 5.3. Now asymptotic relations (5.8)–(5.10) follow from (5.15) and (5.34).
Our first step, Proposition 5.9 below, reduces the asymptotic analysis of and as , to the eigenvalue asymptotics for the Toeplitz operator , . In fact, we formulate Proposition 5.9 in a more general setting.
Letsuch that the operatoris compact. Let. Then for anyandwe haveas.
We omit the standard proof which follows the general lines of [37, Section 5].
Now note that under the hypotheses of Proposition 5.9, the Weyl inequalities (5.14) and the mini-max principle easily imply
for any , , and . Combining (5.35)–(5.36) and (5.37)–(5.38) with , and bearing in mind Corollary 3.7, we obtain
Under the hypotheses of Theorem
5.4
there exists a constantsuch that
Our next goal is to study the asymptotics of and as . To this end, we will apply the approach developed in [13].
Letfor some,, and. Assume that the functionsandsatisfy the condition, andThen there existssuch that
Proposition 5.11 follows from the main theorem of [13] with and , .
Let us now prove the validity of (5.12). By Proposition 5.11,
Since satisfies by assumption condition , we find that
for any and . It is easy to check that our assumption implies the existence of a constant such that
for small enough. Putting together (5.39)–(5.40) and (5.41)–(5.43), we find that there exists a constant such that for any and we have
Choosing and letting , we obtain (5.12). The proof of (5.13) is analogous.
Footnotes
Acknowledgements
The authors are very grateful to Thomas Ransford who gave in [40] the idea of the proof of Proposition . Moreover, they thank Luchezar Stoyanov for a useful discussion of the details of this proof, as well as Dimiter Balabanski and Hajo Leschke for valuable comments on the applications of non-local potentials in nuclear physics. The partial support of the Chilean Science Foundation Fondecyt under Grant 1170816 is gratefully acknowledged.
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. Moreover, they thank Luchezar Stoyanov for a useful discussion of the details of this proof, as well as Dimiter Balabanski and Hajo Leschke for valuable comments on the applications of non-local potentials in nuclear physics. The partial support of the Chilean Science Foundation Fondecyt under Grant 1170816 is gratefully acknowledged.