A Note on the Discrete Spectrum of Gaussian Wells ( I ) : The Ground State Energy in One Dimension

with the latter being regarded as a self-adjoint operator on the domainH(−∞,∞), that is, the second Sobolev space. Its form domain Q(H λ ) is obviously the first Sobolev space. Although the general features of the spectrum of our Hamiltonian are not different from those of the well-known rectangular well, we can take advantage of some special features of the Gaussian attractive potential to determine the ground state ofH λ with great accuracy when λ is small. From the point of view of possible applications, the spectroscopy of such a potential might be of interest in relation to models of “artificial atoms” in the growing field of nanotechnology. For example, in a review article [1], a model of an artificial atom is given by using the two-dimensional harmonic oscillator potential. Although the latter provides a good approximation for the lowest eigenenergies, it is not exactly what we should expect of an atom, due to the absence of the absolutely continuous spectrum. It is therefore interesting to investigate the spectrum of Gaussian wells, since the latter potentials have the typical properties of shortrange potentials but also those of the harmonic oscillator near the bottom of the well.


Introduction
In this brief note, we are concerned with the calculation of the ground state energy of the Hamiltonian: with the latter being regarded as a self-adjoint operator on the domain  2,2 (−∞, ∞), that is, the second Sobolev space.Its form domain (  ) is obviously the first Sobolev space.
Although the general features of the spectrum of our Hamiltonian are not different from those of the well-known rectangular well, we can take advantage of some special features of the Gaussian attractive potential to determine the ground state of   with great accuracy when  is small.
From the point of view of possible applications, the spectroscopy of such a potential might be of interest in relation to models of "artificial atoms" in the growing field of nanotechnology.For example, in a review article [1], a model of an artificial atom is given by using the two-dimensional harmonic oscillator potential.Although the latter provides a good approximation for the lowest eigenenergies, it is not exactly what we should expect of an atom, due to the absence of the absolutely continuous spectrum.It is therefore interesting to investigate the spectrum of Gaussian wells, since the latter potentials have the typical properties of shortrange potentials but also those of the harmonic oscillator near the bottom of the well.

Calculation of the Ground State Energy
As is well known to the mathematical physics community, the Birman-Schwinger kernel is a very useful tool to study the bound states of one-dimensional Hamiltonians with potentials belonging to [2][3][4][5][6]), and also, recently, the work of Fernández [7,8].Used in combination with the Fredholm determinant or similar perturbative arguments, the B-S kernel leads to the well-known approximation for the ground state energy as a function of the coupling constant : for the rather general case − 2 / 2 − , () ≥ 0,  ∈  1 .
We would like to point out that a similar expansion can also be obtained by means of the so-called Titchmarsh-Weyl 2 Advances in Mathematical Physics -function [9][10][11].In our specific case, it is clear that, due to the exponential decay of the Gaussian, we could continue the expansion without limit to get the Taylor series for  0 ().However, that is not what we are going to do in the following, since we are rather interested in an approximation that could give a better evaluation using a smaller number of terms.
Let us check how accurate (2) is in our particular case for a certain value of the coupling constant.Setting  = 0.1 and omitting the error term, we get, after calculating exactly the double integral, leading to It is important to note that the  2 -correction term 0.005 √ 2 is relatively large, roughly, 7 × 10 −3 , which is approximately 8% of the first term 0.05√ in the expansion.This shows that, even for a relatively small value of the coupling constant, it is necessary to include the  2 -correction term to get a good evaluation.Hence, as soon as  increases, more and more terms are required to get a correct evaluation.By using an alternative technique, however, we have found out that it is possible to compute the ground state energy with greater accuracy by determining the root of a transcendental equation.
The crucial step is to use the trace class operator () with integral kernel in momentum space given by in place of the corresponding B-S kernel, following the ideas used in [12].After computing the Fourier transform of our specific potential, the kernel can be rewritten as Using the Taylor expansion of the central exponential, our trace class operator can be rewritten as an infinite sum of rank-one operators; namely, where Obviously, using a well-known result for the explicit calculation of the trace class norm of positive integral operators with continuous kernels (see [13], page 65).Of course, apart from the two subsets of even-and odd-labeled rank-one operators, the operators in the series are not mutually orthogonal.However, it is not difficult, at least conceptually, to transform the sum of all the even-labeled (odd-labeled, resp.)rankone operators into an infinite sum of mutually orthogonal projectors and a sum of nilpotent rank-one operators by constructing an orthogonal system for the functions   (; ).For example, the function  2 (; ) =  2  − 2 /4 /( 2 +  2 ) 1/2 can be expressed as where It is immediate to check that the second summand in ( 10) is orthogonal to the first.
If we are interested in the case of small 's, for which the ground state is the only bound state, a fairly good evaluation of the ground state energy can be obtained by neglecting all the nilpotent operators and taking only the first rankone operator | 0 ()⟩⟨ 0 ()| among the diagonal rank-one operators, since the norm of  0 () is the only one that diverges as  → 0 + .Then, in place of the correct equation det The latter integral has been used in relation to the calculation of the eigenvalues of the Hamiltonian of the harmonic oscillator perturbed by the rational interaction  2 /(1 +  2 ) (see [14][15][16]).Its explicit value, as a function of , is Using such an approximation, the equation for the ground state reads Using Matlab software, the above equation ( 15) is solved for  0 () for different values of ; the ground state energy  0 () is then determined.In the case of  = 0.1, we obtain  0 (0.1) = −6.89617× 10 −3 .This value is more accurate than that previously determined by taking the first two terms in the perturbation expansion of (2).Of course, an even more accurate evaluation can be obtained by including the first order term in the series defining [1 − ()] −1 .Then, the equation reads It would be possible to write (16) almost explicitly by means of ( 6) and ( 9) in [14], where the integrals are thoroughly investigated.However, we have chosen to use a slightly different strategy that enables us to improve the accuracy of our calculations without increasing heavily the mathematical complexity.Essentially, it is sufficient to take only the first two even-labeled rank-one operators in series (7) defining the trace class operator ().As a consequence of ( 10), the second even rank-one operator can be written as where we have omitted the  dependence of the functions to make the notation less heavy and denoted the second summand in (10) by f2 .As we have anticipated, it is reasonable to neglect the two nilpotent operators in (18) since their norm goes like √ and, therefore, is small when  is small.This can be shown as follows: Using the explicit formulae for  2 () and f2 , the latter becomes Solving the above equation using Matlab for various small values of , for which the truncation is satisfactory, leads to the values of  0 () shown in Figure 1 (with some particular values singled out in Table 1) for each value of  in the interval [0, 0.2].Using a Sleign2 algorithm (see [17]) for the calculation of eigenvalues of one-dimensional Schrödinger operators for the case of  = 0.1, a value of  0 (0.1) = −0.006903033148 is determined; the method developed in this paper produces a value of  0 (0.1) = −0.006902786581which is a fairly good agreement.

Table 1 :
The ground state energy for various values of the coupling constant.Although the algebra is a bit lengthy, there is no difficulty in the determination of the new transcendental equation required.We give its final form omitting the intermediate steps: