Structure of the Spectra and Resonances of Schrödinger Operators

The main goal of this paper is to study the spectrum and resonances of several classes of Schrödinger operators. Two important examples occurring in mathematical physics are discussed: harmonic oscillator and Hamiltonian of hydrogen atom.


INTRODUCTION
A big problem in mathematical physics is the description of spectral properties of Schrödinger operators. Here we present some fundamental theorems about spectral and resonance theory of of the operator , which is both accessible to theoretical physicists and chemists and also to researchers in the field of applied mathematics. The essential and discrete spectrum and resonances for such operators have been the subject of a vast literature over the last 20 years, a detailed review and references are found in the book series of Simon (Volume 3, 1979 andVolume 4, 1978) and publications (Aguilar and Combes, 1971;Balslev and Combes, 1971;Brüning, 1989;Muminov and Shodiev, 2011).
We define the discrete spectrum ( ) of a self-adjoint operator as the set of eigenvalues of having finite multiplicity and being isolated points of the spectrum. The essential spectrum of is the set ( ) = ( ) ∖ ( ).
We thus have the disjoint decomposition ( ) = ( ) ∪ ( ). Obviously, ( ) consists of all accumulation points of ( ) and all eigenvalues of infinite multiplicity.
The essential spectrum of a self-adjoint operator is a closed subset of ℝ, it can be characterized as that part of the spectrum that is invariant under compact perturbations, this is the content of Weyl's Theorem, and so, for a Schrödinger operator should morally depend only on properties of the potential .
We also describe the resonances of by using the analytic dilation in a complex strip for suitable potentials (complex Scaling method). Resonances of are complex eigenvalues of the non-hermitian Hamiltonian obtained by complex scaling. Resonance theory of Schrödinger operators based on the complex scaling method is detailed in Messirdi (1994); and Messirdi et al. (2018). However, since the present problem is partly motivated by recent investigations of resonances in periodic structures, we have chosen to investigate resonances Schrödinger operators with periodic potentials.
Note that the problem of computing resonances of with periodic has been much less studied, it is an interesting physical problem with major mathematical difficulties. The resonances of are accessible by meromorphic extension of the resolvent ( − ) −1 , we explain the different steps necessary to get this extension from the upper half plane through the spectrum of . The singularities of ( − ) −1 are effectively the resonances of .
The structure of this paper is as follows. First, in section 2 we study the spectral properties of −∆ on 2 (ℝ ). Section 3, is dedicated to a spectral study of Schrödinger operator in different physical situations where, ( ) → ∞; ( ) → 0 as | | → ∞; is bounded and continuous; is periodic and ( ) = . , ∈ ℝ (Stark-effect), respectively.
Another critical area of mathematical quantum mechanics lies in finding resonances of with a given potential is discussed in section 4 of this paper. We also investigate the situation where the potential is periodic and we compute explicitly the energy spectrum and resonances of hydrogen atom.

SPECTRAL PROPERTIES OF FREE SCHRÖDINGER OPERATOR
Let us first study the spectral properties of −∆ on 2 (ℝ ). We will see later that −∆ does not have eigenvalues. Indeed, the spectrum of −∆ is purely essential covering positive half line.
It is concluded that the operator −∆ on 2 (ℝ ) with domain 2 (ℝ ) does not have eigenvalues. In fact, the spectrum of −∆ is purely continuous.
Proof: It follows from the equation −∆ = that if ∈ 2 (ℝ ), then −∆ ∈ 2 (ℝ ) and so ∈ 4 (ℝ ). Iterating in this manner, it follows that is in any positive indexed Sobolev space. So, the function is infinitely differentiable.

SPECTRAL PROBLEM BACKGROUND FOR SCHRÖDINGER OPERATOR
For Schrödinger operators on 2 (ℝ ), consider as a perturbation of −∆ by the potential Some important spectral results for Schrödinger operators are presented below. We first recall Persson formula for the infimum of the essential spectrum (Persson, 1960).

Lemma:
The following theorems show that under certain conditions on , the essential spectrum of is empty, so the spectrum is purely discrete.

Theorem:
Proof: For any ∈ ℝ, write − = − where ≥ 0 and has compact support. By the Lemma 3.2, is (−Δ + )-compact, so by virtue of perturbation theorem of Weyl we have Since ≥ 0, we know that this is true for all , we conclude that ( ) is empty. The next result is due to Friedrichs (1934).
The associated eigenfunctions form an orthonormal basis of the Hilbert space 2 (ℝ ).

Example:
It is possible to calculate the eigenvalues of explicitly, they are given by , ∈ ℕ * , then: Now consider the sequence ( ) = ( ) . , ∈ ℕ * . Note that for every ∈ ℕ * : Sketch of the proof of Theorem 3.6, or directly thanks to Weyl's theorem, we can prove the theorem for Schrödinger operators with relatively bounded potentials with respect to -Δ and with relative bound < 1. Indeed, first check that ∈ 2 (ℝ 3 ) for all ∈ 2 (ℝ 3 ). Let = ℱ −1 ∈ 2 (ℝ 3 ), so = ℱ where ℱ and ℱ −1 are the Fourier transform and inverse Fourier transform, respectively on 2 (ℝ 3 ). As the functions of 2 (ℝ 3 ) are continuous and tend to zero at infinity, we deduce that is essentially-bounded on ℝ 3 :

Remark:
and, for all , the set { ∈ ℝ ∶ ( ) > } has finite Lebesgue measure. We may now prove a beneficial result, namely that under "small" variations of , the essential spectrum remains the same.

Periodic Potentials
The case where does not have a limit as | | → ∞ in any direction one might expect the analysis of periodic Schrödinger operators to be difficult. Consider a particle moving in a periodic potential. A primary example of this situation is an electron moving in the potential created by ions or atoms of a solid crystal lattice. Linear operators with periodic coefficients arise in many problems both of classical mechanics, mathematics and modern mathematical physic. Such operators appear, e.g. in solid state physics. More details can be found in Cycon et al. (1987). The possible energies of a free electron, corresponding to the free Schrödinger operator −∆ form a straightforward continuum [0, +∞[. The introduction of the periodic perturbation "opens up" gaps in this spectrum. This is called the band structure. In fact the operator = −Δ + can be broken into a family of operators on a n-torus, each having discrete spectrum. More precisely, the spectrum of consists of an increasing sequence of eigenvalues 1 ( ) ≤ 2 ( ) ≤ ⋯ tending to infinity. The manner in which the operators are obtained is not quite unique. Here, we only skim some results for periodic Schrödinger operators. A classical example is the Mathieu operator − 2 2 + cos in 2 (ℝ ), ≠ 0.
In 2 (ℝ 3 ) we study the Schrödinger operator = −∆ + where the potential is a periodic, bounded, real-valued function on ℝ 3 . The periodicity of is defined as follows.
This will permit us to decompose as an hilbertian integral. More precisely, we first define Ω to be the basic lattice cell of the lattice Γ, i.e. the cell spanned by the basis vectors 1 , 2 , 3 of ℝ 3 . Let Γ * be the lattice dual to Γ, i.e. the lattice with basis 1 * , 2 * , 3 * satisfying * ( ) = , the Kronecker symbol, and let Ω * be a basic lattice cell of Γ * . Now for each ∈ Ω * , we define the Hilbert space = 2 (Ω) and we then define the Hilbert space to be the 2) The operator extends uniquely to a unitary operator and 3) The spectrum of is purely discrete for all ∈ Ω * and ( ) = ⋃ ( ) ∈Ω * .
Proof: 1) is automatic by construction.
2) is an isometry on ∞ (ℝ 3 ). Indeed, using Fubini's theorem one has: Hence, ‖ ‖ = ‖ ‖ 2 (ℝ 3 ) and by density extends to an isometry on all of . To show that is in fact a unitary operator, we naturally define * : → by the formula: It's clear that a direct computation shows that * is the adjoint of and is also an isometry, so is a unitary operator and 3 . On the other hand, we know that the spectra of Schrödinger operators on bounded domains with Dirichlet boundary conditions is purely discrete accumulating at +∞ (Cycon et al., 1987).
This is the case of operators , ∈ Ω * , so the spectrum of each , ∈ Ω * , is discrete composed from a sequence of eigenvalues with finite multiplicity 0 ( ) ≤ 1 ( ) ≤ ⋯ ≤ ( ) ≤ ⋯ and lim →+∞ ( ) = +∞. This shows that the spectrum of is union of the sets this is the so-called band structure of spectrum in solid state physics.

Example:
− 2 2 + cos has no eigenvalues, it has purely essential spectrum. It is also shown by Reed and Simon (Volume 4, 1978) that for all ∈ ℕ, +1 ≠ , so every gap occurs.

Stark Potentials
The Hamiltonian describing a quantum mechanical particle in a constant electric field ∈ ℝ is given by ( ) = −∆ + . , where ( ) = . is the Stark potential for the constant electric field . The self-adjoint extension of ( ) on 2 (ℝ ) with its natural domain ( ( )) = 2 (ℝ ) ∩ { ∶ ( . ) ( ) ∈ 2 (ℝ )} is called the Stark-Hamiltonian. We can consider here only the case of one-dimensional electric field. In fact, for example low-dimensional hydrogenic systems are realized in nature in the form of electron-hole pairs ("excitons") in lowdimensional quantum structures such as quantum wells, quantum wires, and carbon nanotubes (Pedersen, 2007). The multi-dimensional case is easily exploited via the spectral properties of self-adjoint operators, in particular the physical case = 3. As ( ( )) ⊆ ℝ, let ∈ ( ( )) and some ∈ ℝ. We want to show that ∈ ( ( )).

RESONANCES OFSCHRÖDINGER OPERATORS
Recently, substantial progress has been given in the analysis of the Schrödinger operator with perturbations going to 0 or ∞ as | | → ∞, or with periodic perturbations, and the works concern the calculation of spectra or resonances of these operators. We now examine the resonances of multi-dimensional Schrödinger operators and we apply our results to those created by the Schrödinger operator for the hydrogen atom.

General Framework
We have studied the discrete and essential spectrum of a self-adjoint operator = −∆ + on of the important physical phenomenon of hydrogen atom (Aguilar and Combes, 1971;Balslev and Combes, 1971;Messirdi, 1994). Let , ∈ ℝ, be the one-parameter group of dilatations: It is easy to check that { ∶ ∈ ℝ} forms a one-parameter unitary group and that ( (−∆)) = 2 (ℝ ) for all ∈ ℝ. Thus, (−Δ) −1 = − −2 Δ is well defined for real , and can be analytically continued into regions of complex . Note that if Im > 0, the spectrum of (−Δ) −1 is −2Im [0 + ∞[, which is the positive half-axis rotated from the origin in the complex plane by an angle equal to (−2Imμ). On the other hand, So, we want to consider a class of analytically continued potentials on a complex strip such that the same is true for the essential spectrum. Assume that is real and the family → ( ) has an analytic continuation into a complex disk {| | < } as operators from the Sobolev space 2 (ℝ ) to 2 (ℝ ), is called a dilation analytic potential. Thus, ( ) is, in this case, a family of non-self-adjoint analytic operators where runs in the disk {| | < }.
Assume also that (−∆ + ) −1 is compact operator, then by Weyl Theorem: Re > ( ) and if there exists small enough, | | < , such that ∈ ( ). It is well known that the resolvent operator ( − ) −1 , ∈ ℂ ∖ ℝ, is a meromorphic function and its poles are precisely the resonances of (Messirdi et al., 2018). Furthermore, the complex poles of ( − ) −1 coincide with the complex poles of the scattering matrix of the system which are interpreted as resonances (Belmouhoub and Messirdi, 2017). The real eigenvalues of ℎ, coincide with eigenvalues of ℎ , complex eigenvalues (resonances) of ℎ, lie in the complex half-plane {Im < 0 ∶ ∈ ℂ} and are locally independent of (see Messirdi et al., 2018).

Resonances for Periodic Schrödinger Operators
We review some results about the resonances for a periodic Schrödinger operator = −∆ + on 2 (ℝ ). If is a real multiplicative potential and periodic with respect to some lattice Γ in ℝ , it was stated in the previous section that the spectrum of consists of bands, these bands consist of purely continuous spectrum. In a neighborhood of such a band, the resonances of are defined and studied by analytic extension of the resolvent ( − ) −1 . It has been shown by Gérard (1990) that ( − ) −1 extends across the spectrum of to the complementary of a discrete set of points, called Van Hove singularities in solid state physics. The existence of such an extension is interesting in the solid state physics to introduce the resonances of the studied system, nevertheless it is mathematically quite difficult to realize. In particular, the singularities of ( − ) −1 are different when we consider the local extension of ( − ) −1 in a small neighborhood of an energy level 0 ∈ ( ), and when we consider the global extension of ( − ) −1 to a bounded open set in ℂ. We will assume that is ∆-bounded with relative bound strictly less than 1, so that is self-adjoint with domain 2 (ℝ ). Using the same notations of the section 3.4, = 2 (Ω * , ; 2 (Ω)) = ∫ ⊕ Ω * where ( , ) and ( , ) are holomorphic for ∈ a bounded set in ℂ such that Ω * ⊂ , and ∈ ℂ, as bounded operators on 2 (Ω).

CONCLUSIONS
The main object of this paper is to study the spectrum of several classes of Schrödinger operators and to look at some important examples occurring in mathematical physics (e.g. the harmonic oscillator, the Stark effect, and the hydrogen atom). We also examine the resonances of multi-dimensional Schrödinger operators and we apply our results to periodic potentials and © CNCS, Mekelle University 102 ISSN: 2220-184X