Conditions for Discreteness of the Spectrum to Schrödinger Operator Via Non-increasing Rearrangement, Lagrangian Relaxation and Perturbations

Abstract

This work is a continuation of our previous paper (Zelenko in Appl Anal Optim 3(2):281–306, 2019), where for the Schrödinger operator $H=-\Delta + V({{\mathbf {x}}})\cdot$, acting in the space $L_2({{\mathbf {R}}}^d)\,(d\ge 3)$, some sufficient conditions for discreteness of its spectrum have been obtained on the base of well known Mazya–Shubin criterion and an optimization problem for a set function. This problem is an infinite-dimensional generalization of a binary linear programming problem. A sufficient condition for discreteness of the spectrum is formulated in terms of the non-increasing rearrangement of the potential $V({{\mathbf {x}}})$. Using the method of Lagrangian relaxation for this optimization problem, we obtain a sufficient condition for discreteness of the spectrum in terms of expectation and deviation of the potential. By means of suitable perturbations of the potential we obtain conditions for discreteness of the spectrum, covering potentials which tend to infinity only on subsets of cubes, whose Lebesgue measures tend to zero when the cubes go to infinity. Also the case where the operator H is defined in the space $L_2(\Omega )$ is considered ($\Omega$ is an open domain in ${{\mathbf {R}}}^d$).

Appendices

Appendix A: Some Classes of Solutions of the Divergence Equation

In this section we consider a class of generalized solutions of the divergence equation

$$\begin{aligned} {\mathrm {div}}\vec \Gamma =W \end{aligned}$$

(A.1)

in the cube $Q_r({{\mathbf {y}}})$. Here W is a distribution on $Q_r({{\mathbf {y}}})$, belonging to

$$\begin{aligned} {\mathcal {D}}^\prime \big ({\mathrm {Int}}(Q_r({{\mathbf {y}}}))\big ). \end{aligned}$$

Recall that the space ${\mathcal {D}}^\prime (\Omega )$ is dual to the topological vector space ${\mathcal {D}}(\Omega )=C_0^\infty (\Omega )$ ($\Omega$ is an open domain in ${{\mathbf {R}}}^d$). Also recall that a vector distribution $\vec \Gamma \in {\mathcal {D}}^\prime \big ({\mathrm {Int}}(Q_r({{\mathbf {y}}})),\,{{\mathbf {R}}}^d\big )$ is called a generalized solution of the above equation, if it satisfies (A.1) in the distributional sense. In particular, this means that if $W\in L_1(Q_r({{\mathbf {y}}}))$ and we consider solutions $\vec \Gamma$, belonging $L_1(Q_r({{\mathbf {y}}}))$, then

$$\begin{aligned}&\forall \,\phi \in C_0^\infty \big ({\mathrm {Int}}(Q_r({{\mathbf {y}}}))\big ):\nonumber \\&\int _{Q_r({{\mathbf {y}}})}W({{\mathbf {x}}})\phi ({{\mathbf {x}}})\, {\mathrm {d}}{{\mathbf {x}}}=-\int _{Q_r({{\mathbf {y}}})}\langle \vec \Gamma ({{\mathbf {x}}}),\,\nabla \phi ({{\mathbf {x}}})\rangle \, {\mathrm {d}}{{\mathbf {x}}}, \end{aligned}$$

(A.2)

([3, 7]). In the case where $W\in {\mathcal {D}}^\prime \big ({{\mathbf {R}}}^d\big )$ and it is 1-periodic in all the variables $x_1,\,x_2,\,\dots ,\,x_d$ we shall consider generalized solutions $\vec \Gamma ({{\mathbf {x}}})$ of (A.1), which are 1-periodic in all the variables. In particular, this means that if $W\in L_{1,\,loc}({{\mathbf {R}}}^d)$ and we consider 1-periodic solutions $\vec \Gamma$, belonging $L_{1,\,loc}({{\mathbf {R}}}^d)$, then

$$\begin{aligned}&\forall \,\phi \in C_0^\infty \big ({{\mathbf {R}}}^d\big ):\nonumber \\&\int _{{{\mathbf {R}}}^d}W({{\mathbf {x}}})\phi ({{\mathbf {x}}})\, {\mathrm {d}}{{\mathbf {x}}}=-\int _{{{\mathbf {R}}}^d}\langle \vec \Gamma ({{\mathbf {x}}}),\,\nabla \phi ({{\mathbf {x}}})\rangle \ {\mathrm {d}}{{\mathbf {x}}}\nonumber \\&{\mathrm {and}}\quad \forall \,\vec l\in {{\mathbf {Z}}}^d\quad \vec \Gamma ({{\mathbf {x}}}+\vec l)=\vec \Gamma ({{\mathbf {x}}}). \end{aligned}$$

(A.3)

1.1 Reduction to ODE

Suppose that $W\in L_1(Q_r({{\mathbf {y}}}))$. We shall look for a solution of Eq. (A.1) of the form:

$$\begin{aligned} \vec \Gamma ({{\mathbf {x}}})=u({{\mathbf {x}}})\,\vec g({{\mathbf {x}}})\quad ({{\mathbf {x}}}=(x_1,\,x_2,\,\dots \,x_d)), \end{aligned}$$

(A.4)

where $\vec g({{\mathbf {x}}})$ is a given vector field on $Q_r({{\mathbf {y}}})\;({{\mathbf {y}}}=(y_1,\,y_2,\,\dots \,y_d))$ and $u({{\mathbf {x}}})$ is a unknown scalar function. Let us take the constant vector field

$$\begin{aligned} \vec g({{\mathbf {x}}})=(1,\,0,\dots ,\,0). \end{aligned}$$

(A.5)

Then (A.1) is reduced to $\frac{\partial u}{\partial x_1}=W({{\mathbf {x}}})$. Consider the solution of last equation in $Q_r({{\mathbf {y}}})$, satisfying the initial condition $u({{\mathbf {x}}})\vert _{x_1=y_1}=0$. It has the form

$$\begin{aligned} u({{\mathbf {x}}})=\int _{y_1}^{x_1}W(s,{{\mathbf {x}}}^\prime )\,{\mathrm {d}}s\quad ({{\mathbf {x}}}^\prime =(x_2,\,x_3,\,\dots ,\,x_d)). \end{aligned}$$

(A.6)

The following claim is valid:

Proposition A.1

Let $\vec \Gamma ({{\mathbf {x}}})$ be the vector field, defined by (A.4), (A.5) and (A.6).

(i) If $W\in C^1(Q_r({{\mathbf {y}}}))$, then $\vec \Gamma ({{\mathbf {x}}})$ is a classical solution of divergence Eq. (A.1) in $Q_r({{\mathbf {y}}})$;

(ii) The linear operator $U(W)({{\mathbf {x}}}):=\int _{y_1}^{x_1}W(s,{{\mathbf {x}}}^\prime )\,{\mathrm {d}}s$, taking part on the right hand side of (A.6) acts in the space $L_p(Q_r({{\mathbf {y}}}))\,(p\ge 1)$ and it is bounded;

(iii) If $W\in L_p(Q_r({{\mathbf {y}}}))$, then $\vec \Gamma ({{\mathbf {x}}})$ is a generalized solution of (A.1), belonging to $L_p(Q_r({{\mathbf {y}}}))$.

Proof

(i) We see from (A.6) that the function $u({{\mathbf {x}}})$ belongs to $C^1(Q_r({{\mathbf {y}}}))$ and it satisfies the equation $\frac{\partial u}{\partial x_1}=W({{\mathbf {x}}})$ in the cube $Q_r({{\mathbf {y}}})$. Hence $\vec \Gamma ({{\mathbf {x}}})$ is a classical solution of (A.1) in $Q_r({{\mathbf {y}}})$. Claim (i) is proven

(ii) We have for $W\in L_p(Q_r({{\mathbf {y}}}))$, using Hölder inequality:

$$\begin{aligned}&\Vert U(W)\Vert _p^p=\int _{Q_r({{\mathbf {y}}})}\Big |\int _{y_1}^{x_1}W(s,{{\mathbf {x}}}^\prime )\,{\mathrm {d}}s\Big |^p\,{\mathrm {d}}{{\mathbf {x}}}\\&\quad \le \int _{Q_r({{\mathbf {y}}})}(x_1-y_1)^{p-1}\int _{y_1}^{x_1}|W(s,{{\mathbf {x}}}^\prime )|^p\,{\mathrm {d}}s\,{\mathrm {d}}x_1\,{\mathrm {d}}{{\mathbf {x}}}^\prime \le \frac{r^p}{p}\Vert W\Vert _p^p. \end{aligned}$$

Claim (ii) is proven.

(iii) By claim (i), for any $W\in C^1(Q_r({{\mathbf {y}}}))$ the vector field $\vec \Gamma ({{\mathbf {x}}})$ is a classical solution of Eq. (A.1), hence it is a generalized solution of it, i.e. the relation (A.2) is valid. On the other hand, in view of (A.4), (A.5), (A.6) and claim (ii) , the correspondence $W\rightarrow \vec \Gamma$ is continuous with respect to $L_p$-norm. Using the density of $C^1(Q_r({{\mathbf {y}}}))$ in $L_p(Q_r({{\mathbf {y}}}))$, we can extend the relation (A.2) by continuity from $W\in C^1(Q_r({{\mathbf {y}}}))$ to $W\in L_p(Q_r({{\mathbf {y}}}))$. We have proved claim (iii). $\square$

1.2 Periodic Potential Solutions

Let us look for solutions of the divergence Eq. (A.1) in the cube $Q_1(\vec 0)$, which satisfy the periodic boundary condition ([3]). This means that we consider these solutions $\vec \Gamma ({{\mathbf {x}}})$ defined on the whole ${{\mathbf {R}}}^d$, which are 1-periodic in all the variables $x_1,\,x_2,\dots ,\,x_d$, i.e. they can be considered as defined on the torus ${{\mathbf {T}}}^d$. Recall that we mean these solutions in the distributional sense (see (A.3)). We shall consider potential solutions of (A.1), i.e. those $\vec \Gamma ({{\mathbf {x}}})$, for which there is a scalar function $\phi ({{\mathbf {x}}})$ such that $\vec \Gamma ({{\mathbf {x}}})=\nabla \phi ({{\mathbf {x}}})$. Then this function should satisfy the Poisson equation

$$\begin{aligned} \Delta \phi =W. \end{aligned}$$

(A.7)

Recall that Fourier operator ${\mathcal {F}}$ on ${{\mathbf {T}}}^d$ maps isometrically the space $L_2({{\mathbf {T}}}^d)$ onto the space $\ell _2({{\mathbf {Z}}}^d)$ [12]. Denote by $L_p^\#({{\mathbf {T}}}^d)$ the set of all functions $f\in L_p({{\mathbf {T}}}^d)$, for which $\int _{{{\mathbf {T}}}^d}f({{\mathbf {x}}})\,{\mathrm {d}}{{\mathbf {x}}}=0$, i.e. ${\mathcal {F}}(f)(\vec 0)=0$. Also denote $\ell _p^0({{\mathbf {Z}}}^d):=\{g\in \ell _p({{\mathbf {Z}}}^d):\;g(\vec 0)=0\}$. We have the following claim:

Proposition A.2

Suppose that $p>2$, $1/p+1/q=1$, $W\in L_2^\#({{\mathbf {T}}}^d)$ and

$$\begin{aligned} \vec G\in \ell _{q}\big ({{\mathbf {Z}}}^d\big ), \end{aligned}$$

(A.8)

where

$$\begin{aligned} \vec G(\vec k)=\frac{{\mathcal {F}}(W)(\vec k)}{2\pi i|\vec k|^2}\vec k. \end{aligned}$$

(A.9)

Then the vector function

$$\begin{aligned} \vec \Gamma ({{\mathbf {x}}})={\mathcal {F}}^{-1}(\vec G)({{\mathbf {x}}}) \end{aligned}$$

(A.10)

belongs to $L_p^\#({{\mathbf {T}}}^d)$ and it is a potential solution of divergence Eq. (A.1). Furthermore,

$$\begin{aligned} |\Vert \vec \Gamma \Vert _p\le d\Vert \vec G\Vert _{\ell _{q}} . \end{aligned}$$

(A.11)

Proof

For brevity we shall omit $({{\mathbf {T}}}^d)$ and $({{\mathbf {Z}}}^d)$ in the notations $L_p({{\mathbf {T}}}^d)$, $L_p^\#({{\mathbf {T}}}^d)$, $l_q({{\mathbf {Z}}}^d)$ and $\ell _q^0({{\mathbf {Z}}}^d)$. Notice that since the operator ${\mathcal {F}}^{-1}:\,\ell _2\rightarrow L_2$ is isometric, for any $g\in \ell _2\;$ $\Vert {\mathcal {F}}^{-1}(g)_2=\Vert g \Vert _{\ell _2}$. On the other hand, in view of the formula for ${\mathcal {F}}^{-1}$, given in Sect. 2, ${\mathcal {F}}^{-1}$ maps $\ell _1$ into $L_\infty$ and for any $g\in \ell _1$ $\Vert {\mathcal {F}}^{-1}(g)\Vert _\infty \le \Vert g \Vert _{\ell _1}$. Then by the Riesz-Thorin Interpolation Theorem ([10], Theorem 1.3.4) , ${\mathcal {F}}^{-1}$ maps $\ell _q$ into $L_p$ and for any $g\in \ell _q$ $\Vert {\mathcal {F}}^{-1}(g)\Vert _p\le \Vert g \Vert _{\ell _q}$. Notice that since $p>2$, $\ell _q\subset \ell _2$ and $L_p\subset L_2$. Then since ${\mathcal {F}}^{-1}:\,\ell _2\rightarrow L_2$ is injective, ${\mathcal {F}}^{-1}:\,\ell _q\rightarrow L_p$ is injective too. It is clear that ${\mathcal {F}}^{-1}$ maps injectively $\ell _q^0$ into $L_p^\#$, It is easy to see that the analogous property is valid for vector functions: ${\mathcal {F}}^{-1}$ maps injectively $\ell _q^0$ into $L_p^\#$ and for any $\vec H\in \ell _q^0$

$$\begin{aligned} \Vert {\mathcal {F}}^{-1}(\vec H)\Vert _p\le d\Vert \vec H\Vert _{\ell _q}. \end{aligned}$$

(A.12)

Applying the Fourier transform to Eq. (A.7) and denoting $u(\vec k)={\mathcal {F}}(\phi )(\vec k)$, we obtain:

$$\begin{aligned} (2\pi i)^2|\vec k|^2u(\vec k)={\mathcal {F}}(W)(\vec k). \end{aligned}$$

Since $W\in L_2^\#$ (hence ${\mathcal {F}}(W)\in \ell _2$ and ${\mathcal {F}}(W)(\vec 0)=0$ ), the last equation has the solution

$$\begin{aligned} u(\vec k)=\frac{{\mathcal {F}}(W)(\vec k)}{(2\pi i)^2|\vec k|^2} . \end{aligned}$$

this solution belongs to the subspace $\ell _2^0$ and it is unique there. Hence the function $\phi ={\mathcal {F}}^{-1}(u)$ is a unique solution of (A.7) in the subspace $L_2^\#$. Moreover, since $|\vec k|^2\ u\in \ell _2$, this solution belongs to the Sobolev space $W_2^2({{\mathbf {T}}}^d)$. Then $\vec \Gamma =\nabla \phi$ is a solution of divergence Eq. (A.1) and it is expressed by (A.10). Furthermore, in view of (A.9), (A.8) and (A.12), inclusion $\vec \Gamma \in L_p^\#$ and estimate (A.11) are valid. $\square$

Appendix B: Some Estimates

Let $\Omega$ be a bounded open domain in ${{\mathbf {R}}}^d$.

Proposition B.1

Suppose that $d>2$.

(i) If $W\in L_{d/2}(\Omega )$, then for any $u\in C_0^\infty (\Omega )$

$$\begin{aligned} |\int _\Omega W({{\mathbf {x}}})|u({{\mathbf {x}}})|^2\, {\mathrm {d}}{{\mathbf {x}}}|\le C^2(d)\Vert W\Vert _{d/2}\int _\Omega |\nabla u({{\mathbf {x}}})|^2\, {\mathrm {d}}{{\mathbf {x}}}, \end{aligned}$$

(B.1)

where C(d) is expressed by (5.11),

(ii) Suppose that there is a vector field $\vec \Gamma ({{\mathbf {x}}})$ on $\Omega$ belonging to $L_d(\Omega )$ and satisfying divergence Eq. (A.1) in the sense of distributions, where $W\in L_1(\Omega )$. Then for any $u\in C_0^\infty (\Omega )$

$$\begin{aligned} |\int _{\Omega } W({{\mathbf {x}}})|u({{\mathbf {x}}})|^2\,{\mathrm {d}}{{\mathbf {x}}}|\le 2\,C(d)\,\Vert \vec \Gamma \Vert _d\int _{\Omega }|\nabla u({{\mathbf {x}}})|^2\, {\mathrm {d}}{{\mathbf {x}}}; \end{aligned}$$

(B.2)

Proof

(i) Using Hölder inequality, we get:

$$\begin{aligned} |\int _\Omega W({{\mathbf {x}}})|u({{\mathbf {x}}})|^2\, {\mathrm {d}}{{\mathbf {x}}}|\le \Vert W\Vert _{d/2}\Vert u\Vert _{q}^2, \end{aligned}$$

where $q=\frac{2d}{d-2}$. On the other hand, by the Sobolev’s theorem, the space $W_2^1({{\mathbf {R}}}^d)$ is embedded continuously into $L_q({{\mathbf {R}}}^d)$. Hence continuing each function $u\in C_0^\infty (\Omega )$ by zero from $\Omega$ to the whole ${{\mathbf {R}}}^d$, we get:

$$\begin{aligned} \Vert u\Vert _{q}^2\le C^2(d)\int _\Omega |\nabla u({{\mathbf {x}}})|^2\, {\mathrm {d}}{{\mathbf {x}}}, \end{aligned}$$

(B.3)

([1, 25]). These circumstances imply the desired claim.

(ii) Making integration by parts, we get for $u\in C_0^\infty (\Omega )$:

$$\begin{aligned} \int _{\Omega }W({{\mathbf {x}}})|u({{\mathbf {x}}})|^2\, {\mathrm {d}}{{\mathbf {x}}}= -2\int _{\Omega }\mathfrak {R}\big (\langle \vec \Gamma {{\mathbf {x}}}),\,\overline{u({{\mathbf {x}}})}\,\nabla u({{\mathbf {x}}})\rangle \big )\, {\mathrm {d}}{{\mathbf {x}}}. \end{aligned}$$

Using Schwartz’s inequality, we obtain:

$$\begin{aligned}&\int _{\Omega }W({{\mathbf {x}}})|u({{\mathbf {x}}})|^2\, {\mathrm {d}}{{\mathbf {x}}}\\&\quad \le 2\,\Vert \vec \Gamma u\Vert _2 \Vert \nabla u\Vert _2. \end{aligned}$$

On the other hand, by claim (i):

$$\begin{aligned} \Vert \vec \Gamma u\Vert _2\le C(d)\Big (\Vert |\vec \Gamma |^2\Vert _{d/2}\Big )^{1/2}\Vert \nabla u\Vert _2.= C(d)\Vert \vec \Gamma \Vert _d\Vert \nabla u\Vert _2. \end{aligned}$$

Te last two estimates imply the desired inequality (B.2). We have proved claim (ii). $\square$