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Low energy spectral and scattering theory

for relativistic Schrödinger operators

###### Abstract

Spectral and scattering theory at low energy for the relativistic Schrödinger operator are investigated. Some striking properties at thresholds of this operator are exhibited, as for example the absence of -energy resonance. Low energy behavior of the wave operators and of the scattering operator are studied, and stationary expressions in terms of generalized eigenfunctions are proved for the former operators. Under slightly stronger conditions on the perturbation the absolute continuity of the spectrum on the positive semi axis is demonstrated. Finally, an explicit formula for the action of the free evolution group is derived. Such a formula, which is well known in the usual Schrödinger case, was apparently not available in the relativistic setting.

Graduate School of Pure and Applied Sciences, University of Tsukuba,

1-1-1 Tennodai, Tsukuba, Ibaraki 305-8571, Japan

E-mail:Department of Mathematical Sciences, University of Hyogo, Shosha,

Himeji 671-2201, Japan

E-mail:

2000 Mathematics Subject Classification: 81U05, 35Q40, 47F05

Keywords: relativistic Schrödinger operators, low energy, scattering theory, wave operators, dilation group

## 1 Introduction

The aim of this paper is to study the spectral and scattering theory of the operator in with a special emphasize on low but positive energies. Various properties of this so-called relativistic Schrödinger operator have already been exhibited in [5, 19, 21], but its corresponding wave operators and scattering operator still deserved investigations. Obviously, the natural comparison operator is the free operator , while for the perturbation it will be assumed that is a measurable real function on satisfying

(1.1) |

for some and almost every . Here, we have used the standard notation .

Now, note that similar investigations for the scattering theory in the usual Schrödinger case (i.e. for the operator ) are part of a piece of folklore. Indeed, based on the seminal work [8], the low energy behavior of the wave operators and of the scattering operator can be derived from stationary expressions for these operators. As for the relativistic Schrödinger operator, on the other hand, the absence of existing information on the behavior of as prevented such a study. For that reason, part of the present work is dedicated to the study of various properties at low energy of the resolvent of the free operator as well as of the perturbed operator. Only once these preliminary results are obtained, further investigations on the scattering theory can be performed.

So, let us be more precise about the framework and about the results. By assuming that satisfies Condition (1.1), then both and are self-adjoint operators with domain equal to the Sobolev space of order on . In addition, the spectrum of consists only of an absolutely continuous part on , while possesses absolutely continuous spectrum on together with a possible discrete set of eigenvalues on which can accumulate only at or at . These results follow from limiting absorption principles which have already been derived in [5].

Now, our first task is the study of the -energy threshold. In particular, one shows that in suitable spaces the operator admits an explicit limit as . Then, one proves that is generically not an eigenvalue for , and that this operator does not possess -energy resonance, see Lemma 2.7 for a precise statement. In the same vein, one also shows that if is not an eigenvalue of , then cannot be an accumulation point of positive eigenvalues of . One should note that such a property has no analog for usual Schrödinger operators. These various spectral results are all derived in Section 2.

Our next task is the derivation of a particular stationary expression for the wave operators ; the definition of can be found at the beginning of Section 3. In fact, such a formula was already announced in [19] but the full proof was lacking. The construction is based on generalized eigenfunctions which can be proved to exist if satisfies Condition (1.1) for . The entire Section 3 is devoted to this proof and the main result expressing the wave operators in terms of generalized eigenfunctions is contained in Proposition 3.4.

Section 4 contains our main new results on the wave operators. Obviously, since can not be diagonalized in the spectral representation of or of , studying the low energy behavior of has to be suitably defined. In fact, our approach relies on the use of the unitary dilation group, which has often been at the root of investigations on rescaled Schrödinger operators, see for example [1]. So, let us recall the action of the dilation group on any , namely for any . Then, the following two relations are of importance, namely and

(1.2) |

where for all . Note that for clarity, the dependence of on both self-adjoint operators used to define them is mentioned. In that setting, our investigations are concentrating on the behavior of the r.h.s. term of (1.2) as . As we shall see in Section 5, this study has a direct consequence on the behavior of the scattering operator at low energy, which is well defined since the scattering operator is diagonal in the spectral representation of .

Now, as already mentioned above, asymptotic properties of can only be derived once suitable information on the resolvent of are obtained. For that purpose, we provide a rather detailed analysis of the operator , with and , as , see Proposition 4.8 where is denoted by . Note that our analysis holds if is not an accumulation point of positive eigenvalues. A comment on this implicit assumption is formulated below. Then, with this information at hand, the main result of Section 4 states that the strong limit is equal to .

The main consequence of this statement concerns the low energy behavior of the scattering operator defined by . In that setting, this corollary states that . Additionally, one also proves a uniform convergence of the scattering operator in the spectral representation of , namely , where is the scattering matrix. This result indicates that there is a significant difference between usual Schrödinger operators and relativistic Schrödinger operators in terms of the low energy asymptotics of the scattering matrices: compare the result of the present paper with the corresponding ones of [8]. What causes this difference is the absence of -energy resonances for relativistic Schrödinger operators. These statements and their proofs correspond to the content of Section 5.

Now, the non-existence of embedded eigenvalues should certainly deserve more attention for the present model. However, since investigations on this question for Schrödinger operators always involve a rather heavy machinery, we do not expect that this question can be easily solved for the present relativistic model. On the other hand, by assuming stronger conditions on , one can rather easily deduce from an abstract argument that the spectrum of on is purely absolutely continuous. Section 6 is devoted to such a result. We clearly suspect that the assumptions on are much too strong for the non-existence of positive eigenvalues, but since the argument is rather simple we have decided to present it for completeness. The proof is based on an abstract result obtained in [16]

Finally, in an appendix, we derive an explicit formula for the action of the unitary propagator . Such a formula, which is well known in the Schrödinger case, was apparently not known in the relativistic case.

In summary, this work contains various results on the low energy behavior of the spectral and the scattering theory of relativistic Schrödinger operators. A similar study for the high energy behavior of these operators would certainly be valuable, and accordingly, a better understanding of the existence or the absence of positive eigenvalues should also deserve some attention. Only once these pre-requisites are fulfilled, a rather complete picture of the scattering theory for relativistic Schrödinger operators would be at hand.

### Notations:

We introduce the notations which will be used in the present paper.

We shall mainly work in the Hilbert space with norm and scalar product denoted by and . Our convention is that the scalar product is linear in its first argument. The weighted Sobolev spaces of order and weight are denoted by . Note that if or is equal to , we simply omit it. A norm on is provided by the expression

where is the position operator and is its conjugate operator in . With these notations, the usual Laplace operator is equal to .

The notation denotes the set of continuous functions on which vanish at infinity. The Schwartz space on is denoted by while defines the set of smooth functions on with compact support.

By extension, for any we denote by the pairing between and , namely for and :

If belongs to and is a tempered distribution, we shall use the notation for their pairing. The usual Fourier transform of is denoted both by and and is defined explicitly on any by

The same notation is used for its standard extension to tempered distributions. As well known, this map is a unitary operator in , and its inverse is denoted by .

For a pair of Hilbert spaces and , denotes the Banach space of all bounded and linear operators from to , and the subset of compact operators. We set for and for .

For complex numbers, we use the standard notation .

## 2 -energy threshold

In this section, we derive various results about the behavior of the resolvent of at . We also provide information about the -energy eigenvalue of and about the absence of -energy resonance for this operator. Finally, we show that if is not an eigenvalue of , then this operator can not have an accumulation of positive eigenvalues at .

We start by studying an auxiliary operator which will be related to the behavior of the resolvent of at . Following [21, Sec. 2], let us set for the operator defined for by

Clearly, this corresponds to the operator of convolution by the function

(2.1) |

It has been shown in [21, Lem. 5.1] that this operator continuously extends to an element of as well as an element of for any . The following statement is a slight improvement of this result.

###### Lemma 2.1.

For any , the operator belongs to and to .

###### Proof.

Let us set for with . Clearly, one has to show that the operators and belong to . We first concentrate on the operator . Recall that for any , . Now, as shown in [21, Eq. (5.6)], can be rewritten as the sum of two functions and with and . Here denotes the characteristic function on the the unit ball in . Thus, one has

where is the Fourier transform of . Then, since and belong to , the product defines a compact operator. Similarly, since and are in , defines a Hilbert-Schmidt operator. Thus, one deduces that the operator is compact.

The similar proof for the operator is omitted. ∎

It clearly follows from this result that belongs to for any . In fact, by real interpolation one also obtains that the operator belongs to for any . Indeed, this result follows from [6] together with the identification of the interpolation spaces , resp. , introduced in that reference with , resp. (see also [2, Sec. 2.8.1] for additional information on real interpolation). In particular, by choosing and , one deduces that belongs to .

Now, it is shown in [5] that the resolvents admit limits as in for any and . In the next statement, we extend this result up to by imposing a stronger condition on the parameter .

###### Lemma 2.2.

For any and , the operators belong to . Furthermore, the maps are continuous in norm and converge to as .

###### Proof.

Recall from [21, Eq. (5.3)] that for any the following formal equalities hold:

(2.2) |

where the definitions of and of are going to be recalled below. Thus, the present proof consists first in introducing the rigorous meaning of (2.2) and then in showing that for the operators and belong to , that they are continuous in norm as functions of , and that they converge in norm to as . Equivalently, one can show the same properties for the operators and in .

It has been proved in [21, Eq. (4.14)] that for any , where are the integral operators defined by

with

(2.3) |

and

(2.4) |

where and are respectively the cosine integral and the sine integral functions. Note that these expressions explicitly define each term in (2.2).

Now, let us observe that . It is well known (see for example [8]) that the map is continuous for and for any with . In particular, this resolvent is continuous as in . Then, by an adequate choice of and , one infers that the maps are continuous in norm and that in norm.

For the compactness of the operator for , let us set for for some . By taking the estimate [21, Eq. (5.16)] into account, namely

(2.5) |

it is easily seen that the function belongs to and thus the operator is a Hilbert-Schmidt operator. Then, let us observe that the relation holds for any . One deduces that

(2.6) |

and that (2.6) vanishes as because of the continuity of the dilation group in . Finally, from the equality one infers that

which implies that .

Clearly, the same estimates and results hold for the operator . Thus, one has obtained that for any , and that the norm of this operator is continuous in and vanishes as when in both norms. By a real interpolation argument, one obtains that the same result holds in . Note that the control on the dependence on for the norm in can be obtained by taking [2, Eq. (2.6.2)] into account. ∎

In Propositions 2.5 and 2.6 below, we show that is generically not an eigenvalue of the operator . To this end, we follow the arguments presented in [3] in the context of Weyl-Dirac operators. For that purpose, we introduce the set as a natural class for the potential . Note that any measurable and real function satisfying Condition (1.1) with belongs to .

###### Lemma 2.3.

If , then is -bounded with relative bound . In particular, is a self-adjoint operator in with domain .

###### Lemma 2.4.

If , then is a compact operator in satisfying

(2.7) |

###### Proof.

We only prove the inequality (2.7), the proof of the compactness can be mimicked directly from the proof of [3, Lem. 1].

To prove the inequality (2.7), we first borrow the Sobolev inequality for from [13, Sec. 8.4] or from [18, p.119, Thm. 1], namely that for any :

(2.8) |

Now let , . We then see that , hence by Lemma 2.3 and by (2.8). Therefore, one can appeal to the definition of the Fourier transform of tempered distributions, and gets

(2.9) |

By applying Hölder inequality twice to (2.9), one obtains

(2.10) |

In the second inequality of (2.10), we have used (2.8). Since is dense in , we find that the inequality (2.10) is valid for all . Hence, it follows that , and the estimate (2.7) is then obtained by density argument. ∎

For the next statements, we need the notation to indicate the dependence on . Let us also denote by the point spectrum of , by the subset and by the dimension of this subset.

###### Proposition 2.5.

Let be in . Then for all except for a discrete subset of .

###### Proof.

Let us define a -valued analytic function on by

By Lemma 2.4, is a compact operator for each . Therefore, one can apply the analytic Fredholm theorem (see for example [15, p. 201]) and deduce that is invertible in for all except for a discrete subset of . In particular, one infers that for all except for a discrete subset of .

Now, let such that , and let us assume that there exists . Clearly, one has . Then, let us set which satisfies

(2.11) |

It is obvious that (2.11) is equivalent to . This implies that , because is invertible in . Since is an injective mapping from to , it follows that , and we can conclude that whenever is invertible in . ∎

###### Proposition 2.6.

The set contains an open and dense subset of .

###### Proof.

Let us now set

for any . In the sequel, we show that the set defined by

is open and dense in . Then, the statement of the Proposition is a consequence of the inclusion which has already been proved in the second half of the previous proof.

Let . Since is a compact operator in by Lemma 2.4, we observe that is invertible in . Now choose a real number small enough such that . If satisfies , then the identity

together with (2.7), enables one to construct the inverse of by a Neumann series. Hence, , and then is an open subset of .

To prove the density of , let and be given. It then follows from the proof of Proposition 2.5 that is invertible in for all except for a discrete subset of . This means that one can choose so that and that . Therefore , and then is dense in . ∎

We now derive two results about the absence of -energy resonance. Before this, we recall from [20, Thm. 4.6] that if for some , then is a tempered distribution.

###### Lemma 2.7.

Assume that satisfies Condition (1.1) with . Assume that there exists for some satisfying in the sense of distributions. Then , i.e. is a -energy eigenfunction.

###### Proof.

By assumption, the equality holds in the sense of distributions. Since the r.h.s. belongs to with , this already implies that . In addition, it follows from the regularity properties of that , see Lemma 2.1 and the paragraph before it. Now, let and observe that

Consequently, , and by density, it follows that in . In conclusion, both and belong to . Since the domain of is , it follows that . ∎

We remark that -energy resonances are often understood as states behaving at infinity in a prescribed way, as described by (2.12) below (cf. [14, p. 123505-5], [17, Sec. 6], [23, Sec. 1]). We can also show the absence of such -energy resonances if .

###### Lemma 2.8.

Assume that satisfies Condition (1.1) with . Suppose also that there exist and such that in the sense of distributions and that

(2.12) |

where the convergence is uniform on as . Then and .

###### Proof.

It is easy to see that there exist two constants and such that for all with . This fact, together with the assumption, implies that for any . Since , it follows from Lemma 2.7 that .

We prove that by contradiction. To this end, we suppose that . Then there exist a constant and a measurable subset of such that and on . This easily implies that the first term on the right hand side of (2.12) is not square-integrable on the set , while the second and third terms are square-integrable on the same set. Since , this is a contradiction. ∎

We conclude this section with a theorem, which asserts that if is not an eigenvalue of , then cannot be an accumulation point of positive eigenvalues of .

###### Theorem 2.9.

Assume that satisfies Condition (1.1) with , and that . Then there exists a constant such that .

To prove this assertion, we need two preliminary lemmas.

###### Lemma 2.10.

Assume that satisfies Condition (1.1) with , and that . Then, the operator is invertible in for any .

###### Proof.

In view of Lemma 2.2, is a compact operator in for any . Therefore, it is sufficient to show that is not an eigenvalue of .

Let satisfy . Here we may assume, without loss of generality, that is sufficiently close to . Putting , we find that and in the sense of distributions. It follows that in the sense of distributions. To apply Lemma 2.7, we note that , hence that for some . Thus we can conclude from Lemma 2.7 that . This implies that , because by assumption of the lemma. Finally, we see that . ∎

###### Lemma 2.11.

Assume that satisfies Condition (1.1) with . Let and let satisfy . Then .

###### Proof.

###### Proof of Theorem 2.9.

Let us first recall that in any unital Banach algebra, the set of invertible elements is an open set. Then, for each fixed , it follows from Lemmas 2.2 and 2.10 that there exists a positive constant such that for each , the operators are invertible in .

To prove the proposition, let and suppose that satisfies . Then by Lemma 2.11, we find that with . This means that , or in other words that . Since , it follows from the previous paragraph that , and then . We have thus shown that . ∎

## 3 Stationary expression for the wave operators

In this section we derive stationary expressions for the wave operators which were already announced in [19]. Since the limiting absorption principle and the generalized eigenfunction expansions for the operator was established in [21], we can follow the line of [11, Sec. 2], where the discussions were made in an abstract setting, and the line of [12, Chapt. 5], where the discussions were given for the three-dimensional Schrödinger operator.

For , let us recall that and are used respectively for the resolvents and . The notation is used for the spectral measure of . We also recall that the following limiting absorption principle has been proved in [5], namely for and the operators belong to . Note that the condition in (1.1) has been tacitly assumed. As a consequence, the wave operators defined by the strong limits

exist and are asymptotically complete. In addition, these expressions are equal to the ones obtained by the usual stationary approach, see for example [22, Thm. 5.3.6].

###### Lemma 3.1.

Let and assume that , belong to with for some . Then one has

(3.1) |

where .

Note that it follows from the hypothesis on that for any Borel set , and that for all . Thus, the usual integral over reduces to an integral over the finite interval . The following proof is standard, but we recall it for completeness.